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|---|---|---|---|---|---|---|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
|
refine' ⟨_, this⟩
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
|
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω = hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) M ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
|
refine' hitting_eq_hitting_of_exists hNM _
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
|
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : ∃ j ∈ Set.Ico (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
case hi
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ N ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
|
obtain ⟨j, hj₁, hj₂⟩ := h
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case intro.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
j : ℕ
hj₁ : j ∈ Set.Ico (lowerCrossingTime a b f N n ω) N
hj₂ : f j ω ∈ Set.Ici b
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
case hi
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ N ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
|
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
|
Mathlib.Probability.Martingale.Upcrossing.521_0.80Cpy4Qgm9i1y9y
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N n ω < N
⊢ upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
|
cases n
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
|
Mathlib.Probability.Martingale.Upcrossing.535_0.80Cpy4Qgm9i1y9y
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case zero
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N Nat.zero ω < N
⊢ upperCrossingTime a b f M Nat.zero ω = upperCrossingTime a b f N Nat.zero ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
·
|
simp
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
·
|
Mathlib.Probability.Martingale.Upcrossing.535_0.80Cpy4Qgm9i1y9y
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case succ
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
M : ℕ
hNM : N ≤ M
n✝ : ℕ
h : upperCrossingTime a b f N (Nat.succ n✝) ω < N
⊢ upperCrossingTime a b f M (Nat.succ n✝) ω = upperCrossingTime a b f N (Nat.succ n✝) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
·
|
exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
·
|
Mathlib.Probability.Martingale.Upcrossing.535_0.80Cpy4Qgm9i1y9y
|
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
⊢ Monotone fun N ω => upcrossingsBefore a b f N ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
|
intro N M hNM ω
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
⊢ (fun N ω => upcrossingsBefore a b f N ω) N ω ≤ (fun N ω => upcrossingsBefore a b f N ω) M ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
|
simp only [upcrossingsBefore]
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
⊢ sSup {n | upperCrossingTime a b f N n ω < N} ≤ sSup {n | upperCrossingTime a b f M n ω < M}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
|
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : Set.Nonempty {n | upperCrossingTime a b f N n ω < N}
⊢ sSup {n | upperCrossingTime a b f N n ω < N} ≤ sSup {n | upperCrossingTime a b f M n ω < M}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
·
|
refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
·
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n✝ m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : Set.Nonempty {n | upperCrossingTime a b f N n ω < N}
n : ℕ
hn : n ∈ {n | upperCrossingTime a b f N n ω < N}
⊢ n ∈ {n | upperCrossingTime a b f M n ω < M}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
|
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n✝ m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : Set.Nonempty {n | upperCrossingTime a b f N n ω < N}
n : ℕ
hn : n ∈ {n | upperCrossingTime a b f N n ω < N}
⊢ upperCrossingTime a b f N n ω < M
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
|
exact lt_of_lt_of_le hn hNM
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : ¬Set.Nonempty {n | upperCrossingTime a b f N n ω < N}
⊢ sSup {n | upperCrossingTime a b f N n ω < N} ≤ sSup {n | upperCrossingTime a b f M n ω < M}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
·
|
rw [Set.not_nonempty_iff_eq_empty] at hemp
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
·
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : {n | upperCrossingTime a b f N n ω < N} = ∅
⊢ sSup {n | upperCrossingTime a b f N n ω < N} ≤ sSup {n | upperCrossingTime a b f M n ω < M}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
|
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
|
Mathlib.Probability.Martingale.Upcrossing.543_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
|
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ Nat.succ (upcrossingsBefore a b f N ω) ∈ {n | upperCrossingTime a b f (N₂ + 1) n ω < N₂ + 1}
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
|
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ ∃ j ∈ Set.Ico (lowerCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω) (N₂ + 1), f j ω ∈ Set.Ici b
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
·
|
refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
·
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ lowerCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N₂
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
|
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ ∃ j ∈ Set.Icc (upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω) N₂, f j ω ∈ Set.Iic a
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
|
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
|
by_cases hN : 0 < N
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : 0 < N
⊢ upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
·
|
have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
·
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : 0 < N
this : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N
⊢ upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
|
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : 0 < N
this : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N
⊢ upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
|
exact this.le
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : ¬0 < N
⊢ upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
·
|
rw [not_lt, le_zero_iff] at hN
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
·
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : N = 0
⊢ upperCrossingTime a b f (N₂ + 1) (upcrossingsBefore a b f N ω) ω ≤ N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
|
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
hN : N = 0
⊢ ⊥ ω ≤ 0
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
|
rfl
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
|
Mathlib.Probability.Martingale.Upcrossing.554_0.80Cpy4Qgm9i1y9y
|
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω - stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
|
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
|
Mathlib.Probability.Martingale.Upcrossing.588_0.80Cpy4Qgm9i1y9y
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ N ≤ upperCrossingTime a b f N n ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
|
rw [upcrossingsBefore] at hn
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
|
Mathlib.Probability.Martingale.Upcrossing.588_0.80Cpy4Qgm9i1y9y
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hn : sSup {n | upperCrossingTime a b f N n ω < N} < n
⊢ N ≤ upperCrossingTime a b f N n ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
|
rw [← not_lt]
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
|
Mathlib.Probability.Martingale.Upcrossing.588_0.80Cpy4Qgm9i1y9y
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hn : sSup {n | upperCrossingTime a b f N n ω < N} < n
⊢ ¬upperCrossingTime a b f N n ω < N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
|
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
|
Mathlib.Probability.Martingale.Upcrossing.588_0.80Cpy4Qgm9i1y9y
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hn : upcrossingsBefore a b f N ω < n
this : N ≤ upperCrossingTime a b f N n ω
⊢ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω - stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
|
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
|
Mathlib.Probability.Martingale.Upcrossing.588_0.80Cpy4Qgm9i1y9y
|
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
|
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
|
by_cases hN : N = 0
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
·
|
simp [hN]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
|
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
∑ x in Finset.range N,
∑ x_1 in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N x_1 ω) (upperCrossingTime a b f N (x_1 + 1) ω))
(fun a => f (a + 1) ω - f a ω) x
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
|
rw [Finset.sum_comm]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
∑ y in Finset.range N,
∑ x in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N y ω) (upperCrossingTime a b f N (y + 1) ω))
(fun a => f (a + 1) ω - f a ω) x
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
|
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ ∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
|
intro k
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k : ℕ
⊢ ∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
|
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k : ℕ
⊢ f (upperCrossingTime a b f N (k + 1) ω) ω - f (lowerCrossingTime a b f N k ω) ω =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
·
|
rfl
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k : ℕ
⊢ Finset.filter (fun i => i ∈ Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
·
|
ext i
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case a
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k i : ℕ
⊢ i ∈
Finset.filter (fun i => i ∈ Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(Finset.range N) ↔
i ∈ Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
|
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case a
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
k i : ℕ
⊢ lowerCrossingTime a b f N k ω ≤ i → i < upperCrossingTime a b f N (k + 1) ω → i < N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
|
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
∑ y in Finset.range N,
∑ x in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N y ω) (upperCrossingTime a b f N (y + 1) ω))
(fun a => f (a + 1) ω - f a ω) x
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
|
simp_rw [h₁]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
∑ x in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (x + 1)) ω - stoppedValue f (lowerCrossingTime a b f N x) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
|
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
⊢ ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
⊢ ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
hi : i ∈ Finset.range (upcrossingsBefore a b f N ω)
⊢ i < upcrossingsBefore a b f N ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
|
rwa [Finset.mem_range] at hi
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
⊢ ∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω) ≤
∑ k in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : i ∉ Finset.range (upcrossingsBefore a b f N ω)
⊢ 0 ≤ stoppedValue f (upperCrossingTime a b f N (i + 1)) ω - stoppedValue f (lowerCrossingTime a b f N i) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
|
by_cases hi' : i = upcrossingsBefore a b f N ω
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : i ∉ Finset.range (upcrossingsBefore a b f N ω)
hi' : i = upcrossingsBefore a b f N ω
⊢ 0 ≤ stoppedValue f (upperCrossingTime a b f N (i + 1)) ω - stoppedValue f (lowerCrossingTime a b f N i) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
·
|
subst hi'
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
⊢ 0 ≤
stoppedValue f (upperCrossingTime a b f N (upcrossingsBefore a b f N ω + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω)) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
|
simp only [stoppedValue]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
⊢ 0 ≤
f (upperCrossingTime a b f N (upcrossingsBefore a b f N ω + 1) ω) ω -
f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
|
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
⊢ 0 ≤ f N ω - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
|
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
⊢ 0 ≤ f N ω - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
·
|
rw [heq, sub_self]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
heq : ¬lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
⊢ 0 ≤ f N ω - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
·
|
rw [sub_nonneg]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
x✝ : upcrossingsBefore a b f N ω ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ∉ Finset.range (upcrossingsBefore a b f N ω)
heq : ¬lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
⊢ f (lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω) ω ≤ f N ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
|
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : i ∉ Finset.range (upcrossingsBefore a b f N ω)
hi' : ¬i = upcrossingsBefore a b f N ω
⊢ 0 ≤ stoppedValue f (upperCrossingTime a b f N (i + 1)) ω - stoppedValue f (lowerCrossingTime a b f N i) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
·
|
rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
·
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : i ∉ Finset.range (upcrossingsBefore a b f N ω)
hi' : ¬i = upcrossingsBefore a b f N ω
⊢ upcrossingsBefore a b f N ω < i
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
|
rw [Finset.mem_range, not_lt] at hi
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ≤ i
hi' : ¬i = upcrossingsBefore a b f N ω
⊢ upcrossingsBefore a b f N ω < i
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
|
exact lt_of_le_of_ne hi (Ne.symm hi')
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
h₂ :
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω)
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
∑ x in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (x + 1)) ω - stoppedValue f (lowerCrossingTime a b f N x) ω)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
|
refine' le_trans _ h₂
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n in Finset.range N,
Set.indicator (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω))
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
h₂ :
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N,
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω)
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
|
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
|
Mathlib.Probability.Martingale.Upcrossing.599_0.80Cpy4Qgm9i1y9y
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hfN : ∀ (ω : Ω), a ≤ f N ω
hfzero : 0 ≤ f 0
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤
∫ (x : Ω), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
|
rw [← integral_mul_left]
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
|
Mathlib.Probability.Martingale.Upcrossing.655_0.80Cpy4Qgm9i1y9y
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N]
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hfN : ∀ (ω : Ω), a ≤ f N ω
hfzero : 0 ≤ f 0
hab : a < b
⊢ ∫ (a_1 : Ω), (b - a) * ↑(upcrossingsBefore a b f N a_1) ∂μ ≤
∫ (x : Ω), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
|
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
|
Mathlib.Probability.Martingale.Upcrossing.655_0.80Cpy4Qgm9i1y9y
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N]
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_1
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hfN : ∀ (ω : Ω), a ≤ f N ω
hfzero : 0 ≤ f 0
hab : a < b
⊢ 0 ≤ᵐ[μ] fun a_1 => (b - a) * ↑(upcrossingsBefore a b f N a_1)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
·
|
exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
·
|
Mathlib.Probability.Martingale.Upcrossing.655_0.80Cpy4Qgm9i1y9y
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N]
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_2
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hfN : ∀ (ω : Ω), a ≤ f N ω
hfzero : 0 ≤ f 0
hab : a < b
⊢ (fun a_1 => (b - a) * ↑(upcrossingsBefore a b f N a_1)) ≤ᵐ[μ] fun x =>
Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
·
|
refine' eventually_of_forall fun ω => _
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
·
|
Mathlib.Probability.Martingale.Upcrossing.655_0.80Cpy4Qgm9i1y9y
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N]
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_2
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hfN : ∀ (ω : Ω), a ≤ f N ω
hfzero : 0 ≤ f 0
hab : a < b
ω : Ω
⊢ (fun a_1 => (b - a) * ↑(upcrossingsBefore a b f N a_1)) ω ≤
(fun x => Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
|
simpa using mul_upcrossingsBefore_le (hfN ω) hab
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
|
Mathlib.Probability.Martingale.Upcrossing.655_0.80Cpy4Qgm9i1y9y
|
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N]
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
|
have hab' : 0 < b - a := sub_pos.2 hab
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
|
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
⊢ ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
|
intro i ω
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
⊢ b - a ≤ (f ω i - a)⁺ ↔ b ≤ f ω i
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
|
refine' ⟨fun h => _, fun h => _⟩
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_1
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b - a ≤ (f ω i - a)⁺
⊢ b ≤ f ω i
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
·
|
rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_2
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b ≤ f ω i
⊢ b - a ≤ (f ω i - a)⁺
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
·
|
rw [← sub_le_sub_iff_right a] at h
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case refine'_2
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b - a ≤ f ω i - a
⊢ b - a ≤ (f ω i - a)⁺
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
|
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
|
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
⊢ ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
|
intro ω i
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
ω : Ω
i : ℕ
⊢ (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
|
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
|
induction' n with k ih
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case zero
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N Nat.zero = upperCrossingTime a b f N Nat.zero ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N Nat.zero = lowerCrossingTime a b f N Nat.zero
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
·
|
refine' ⟨rfl, _⟩
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case zero
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
⊢ lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N Nat.zero = lowerCrossingTime a b f N Nat.zero
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
|
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case zero
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
⊢ (fun x => if ∃ j ∈ Set.Icc ⊥ N, (f j x - a)⁺ ∈ Set.Iic 0 then sInf (Set.Icc ⊥ N ∩ {i | (f i x - a)⁺ ≤ 0}) else N) =
fun x => if ∃ j ∈ Set.Icc ⊥ N, f j x ∈ Set.Iic a then sInf (Set.Icc ⊥ N ∩ {i | f i x ≤ a}) else N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
|
ext ω
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case zero.h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
⊢ (if ∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0 then sInf (Set.Icc ⊥ N ∩ {i | (f i ω - a)⁺ ≤ 0}) else N) =
if ∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a then sInf (Set.Icc ⊥ N ∩ {i | f i ω ≤ a}) else N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
|
split_ifs with h₁ h₂ h₂
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₁ : ∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a
⊢ sInf (Set.Icc ⊥ N ∩ {i | (f i ω - a)⁺ ≤ 0}) = sInf (Set.Icc ⊥ N ∩ {i | f i ω ≤ a})
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
·
|
simp_rw [hf']
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₁ : ∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ¬∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a
⊢ sInf (Set.Icc ⊥ N ∩ {i | (f i ω - a)⁺ ≤ 0}) = N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
·
|
simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₁ : ∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ¬∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ≤ 0
⊢ sInf (Set.Icc ⊥ N ∩ {i | (f i ω - a)⁺ ≤ 0}) = N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
|
exact False.elim (h₂ h₁)
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₁ : ¬∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a
⊢ N = sInf (Set.Icc ⊥ N ∩ {i | f i ω ≤ a})
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
·
|
simp_rw [Set.mem_Iic, hf' _ _] at h₁
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₂ : ∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a
h₁ : ¬∃ j ∈ Set.Icc ⊥ N, f j ω ≤ a
⊢ N = sInf (Set.Icc ⊥ N ∩ {i | f i ω ≤ a})
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
|
exact False.elim (h₁ h₂)
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
ω : Ω
h₁ : ¬∃ j ∈ Set.Icc ⊥ N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ¬∃ j ∈ Set.Icc ⊥ N, f j ω ∈ Set.Iic a
⊢ N = N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
·
|
rfl
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case succ
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (Nat.succ k) = upperCrossingTime a b f N (Nat.succ k) ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
·
|
have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
simp_all only [Set.mem_Ici, not_true_eq_false]
· refine' False.elim (h₁ _)
simp_all only [Set.mem_Ici]
· rfl
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) = upperCrossingTime a b f N (k + 1)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
|
ext ω
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) ω = upperCrossingTime a b f N (k + 1) ω
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
|
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
⊢ (if ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a) then
sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ (f i ω - a)⁺ + a})
else N) =
if ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b then
sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ f i ω})
else N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
|
split_ifs with h₁ h₂ h₂
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ (f i ω - a)⁺ + a}) =
sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ f i ω})
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
·
|
simp_rw [← sub_le_iff_le_add, hf ω]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ¬∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ (f i ω - a)⁺ + a}) = N
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
·
|
refine' False.elim (h₂ _)
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ¬∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
|
simp_all only [Set.mem_Ici, not_true_eq_false]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ¬∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ N = sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ f i ω})
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
simp_all only [Set.mem_Ici, not_true_eq_false]
·
|
refine' False.elim (h₁ _)
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
simp_all only [Set.mem_Ici, not_true_eq_false]
·
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ¬∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
/-!
# Doob's upcrossing estimate
Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the
number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing
estimate (also known as Doob's inequality) states that
$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$
Doob's upcrossing estimate is an important inequality and is central in proving the martingale
convergence theorems.
## Main definitions
* `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f`
crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is
taken to be `N`).
* `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is
between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively
one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process
crosses below `a` for the first time after selling and selling 1 share whenever the process
crosses above `b` for the first time after buying.
* `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to
above `b` before time `N`.
* `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above
`b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`.
## Main results
* `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a
stopping time whenever the process it is associated to is adapted.
* `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's
upcrossing estimate.
* `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality
obtained by taking the supremum on both sides of Doob's upcrossing estimate.
### References
We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021]
-/
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators Topology
namespace MeasureTheory
variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
/-!
## Proof outline
In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$
to above $b$ before time $N$.
To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses
below $a$ and above $b$. Namely, we define
$$
\sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N;
$$
$$
\tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N.
$$
These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined
using `MeasureTheory.hitting` allowing us to specify a starting and ending time.
Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$.
Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that
$0 \le f_0$ and $a \le f_N$. In particular, we will show
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N].
$$
This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization.
To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$
(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is
a submartingale if $(f_n)$ is.
Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that
$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$,
$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property,
$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying
$$
\mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0].
$$
Furthermore,
\begin{align}
(C \bullet f)_N & =
\sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\
& = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1}
+ \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\
& = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k})
\ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b)
\end{align}
where the inequality follows since for all $k < U_N(a, b)$,
$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$,
$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and
$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have
$$
(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N]
\le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N],
$$
as required.
To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$.
-/
/-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. -/
noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) :
Ω → ι :=
hitting f (Set.Iic a) c N
#align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux
/-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. -/
noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) : ℕ → Ω → ι
| 0 => ⊥
| n + 1 => fun ω =>
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω
#align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime
/-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. -/
noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω
#align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime
section
variable [Preorder ι] [OrderBot ι] [InfSet ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ :=
rfl
#align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N :=
rfl
#align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
rw [upperCrossingTime]
#align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
rfl
#align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq
end
section ConditionallyCompleteLinearOrderBot
variable [ConditionallyCompleteLinearOrderBot ι]
variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω}
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
#align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ :=
eq_bot_iff.2 upperCrossingTime_le
#align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero'
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
#align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le
theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω]
#align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
#align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime
#align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
#align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono
end ConditionallyCompleteLinearOrderBot
variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω}
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩
#align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
#align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn)
simp only [stoppedValue]
rw [← h]
exact stoppedValue_upperCrossingTime (h.symm ▸ hn)
#align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
(hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>
not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) _
simp only [stoppedValue]
rw [← h]
exact stoppedValue_lowerCrossingTime (h.symm ▸ hn)
#align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_lt_upperCrossingTime hab hn)
#align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
lowerCrossingTime a b f N m ω = N :=
le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm))
#align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
upperCrossingTime a b f N m ω = N :=
le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm))
#align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
lowerCrossingTime a b f N m ω = N :=
lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn)
#align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
upperCrossingTime a b f N m ω = N :=
upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn)
#align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize'
-- `upperCrossingTime_bound_eq` provides an explicit bound
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upperCrossingTime a b f N n ω = N := by
by_contra h; push_neg at h
have : StrictMono fun n => upperCrossingTime a b f N n ω :=
strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _)
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m :=
⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩,
lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩
exact not_le.2 hk upperCrossingTime_le
#align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine' ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => _⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
#align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
{n | upperCrossingTime a b f N n ω < N}.Nonempty :=
⟨0, hN⟩
#align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upperCrossingTime a b f N N ω = N := by
by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab)
· refine' le_antisymm upperCrossingTime_le _
have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω)
(Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by
refine' strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab _
rw [Nat.lt_pred_iff] at hm
convert Nat.find_min _ hm
convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN')
· rw [not_lt] at hN'
exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab))
#align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upperCrossingTime a b f N n ω = N :=
le_antisymm upperCrossingTime_le
(le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn))
#align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le
variable {ℱ : Filtration ℕ m0}
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by
induction' n with k ih
· refine' ⟨isStoppingTime_const _ 0, _⟩
simp [hitting_isStoppingTime hf measurableSet_Iic]
· obtain ⟨_, ih₂⟩ := ih
have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by
intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine' ⟨this, _⟩
· intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _
#align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (upperCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.1
#align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
IsStoppingTime ℱ (lowerCrossingTime a b f N n) :=
hf.isStoppingTime_crossing.2
#align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
/-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted
rather than predictable. -/
noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ :=
∑ k in Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n
#align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω :=
Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _
#align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by
rw [upcrossingStrat, ← Finset.indicator_biUnion_apply]
· exact Set.indicator_le_self' (fun _ _ => zero_le_one) _
intro i _ j _ hij
simp only [Set.Ico_disjoint_Ico]
obtain hij' | hij' := lt_or_gt_of_ne hij
· rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_right (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
· rw [gt_iff_lt] at hij'
rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :
upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω),
max_eq_left (lowerCrossingTime_mono hij'.le :
lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)]
refine' le_trans upperCrossingTime_le_lowerCrossingTime
(lowerCrossingTime_mono (Nat.succ_le_of_lt hij'))
#align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
Adapted ℱ (upcrossingStrat a b f N) := by
intro n
change StronglyMeasurable[ℱ n] fun ω =>
∑ k in Finset.range N, ({n | lowerCrossingTime a b f N k ω ≤ n} ∩
{n | n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n
refine' Finset.stronglyMeasurable_sum _ fun i _ =>
stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter _)
simp_rw [← not_le]
exact (hf.isStoppingTime_upperCrossingTime n).compl
#align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ :=
hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ =>
upcrossingStrat_nonneg
#align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by
refine' hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n))
(_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) _
· exact fun n ω => sub_le_self _ upcrossingStrat_nonneg
· intro n ω
simp [upcrossingStrat_le_one]
#align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by
have h₁ : (0 : ℝ) ≤
μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).set_integral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine' le_trans _ this
simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl]
have h₂ : μ[∑ k in Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] =
μ[∑ k in Finset.range n, (f (k + 1) - f k)] -
μ[∑ k in Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply,
Pi.mul_apply]
refine' integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _)
(integrable_finset_sum _ fun i _ => hf.integrable _)) _
convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1
ext; simp
rw [h₂, sub_nonneg] at h₁
refine' le_trans h₁ _
simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl]
#align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
/-- The number of upcrossings (strictly) before time `N`. -/
noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ)
(N : ι) (ω : Ω) : ℕ :=
sSup {n | upperCrossingTime a b f N n ω < N}
#align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
{ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore]
#align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by
ext ω; exact upcrossingsBefore_zero
#align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero'
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N :=
haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
(upperCrossingTime_lt_nonempty hN).cSup_mem
((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab))
lt_of_le_of_lt (upperCrossingTime_mono hn) this
#align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
(hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
refine' le_antisymm upperCrossingTime_le (not_lt.1 _)
convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
#align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossingsBefore a b f N ω ≤ N := by
by_cases hN : N = 0
· subst hN
rw [upcrossingsBefore_zero]
· refine' csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => _
by_contra hnN
exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le)
#align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : lowerCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have h' : upperCrossingTime a b f N n ω < N :=
lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
induction' n with k ih
· simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
lowerCrossingTime_zero, true_and_iff, eq_comm]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
· specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h)
(lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h')
have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h'
simp only [upperCrossingTime_succ_eq]
obtain ⟨j, hj₁, hj₂⟩ := h'
rw [eq_comm, ih.2]
exacts [hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
refine' ⟨this, _⟩
simp only [lowerCrossingTime, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩
#align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by
have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine' ⟨_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine' hitting_eq_hitting_of_exists hNM _
rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h
obtain ⟨j, hj₁, hj₂⟩ := h
exacts [⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩, le_rfl]
#align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N n ω < N) :
upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by
cases n
· simp
· exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1
#align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine' csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => _
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· rw [Set.not_nonempty_iff_eq_empty] at hemp
simp [hemp, csSup_empty, bot_eq_zero', zero_le']
#align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by
refine' lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) _)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
· refine' ⟨N₂, ⟨_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine' ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩
by_cases hN : 0 < N
· have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N :=
Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab)
rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <| Nat.le_succ _))
this]
exact this.le
· rw [not_lt, le_zero_iff] at hN
rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero]
rfl
#align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
#align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) :
b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω :=
sub_le_sub
(stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne)
(stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne)
#align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by
have : N ≤ upperCrossingTime a b f N n ω := by
rw [upcrossingsBefore] at hn
rw [← not_lt]
exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h)
simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,
lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)]
#align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k in Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by
classical
by_cases hN : N = 0
· simp [hN]
simp_rw [upcrossingStrat, Finset.sum_mul, ←
Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul]
rw [Finset.sum_comm]
have h₁ : ∀ k, ∑ n in Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω)
(upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω := by
intro k
rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico
(lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) =
Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)),
Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,
Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub,
sub_add_sub_cancel]
· rfl
· ext i
simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,
and_iff_right_iff_imp, and_imp]
exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le
simp_rw [h₁]
have h₂ : ∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
calc
∑ _k in Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k in Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum fun i hi =>
le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _
rwa [Finset.mem_range] at hi
_ ≤ ∑ k in Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω -
stoppedValue f (lowerCrossingTime a b f N k) ω) := by
refine' Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => _
by_cases hi' : i = upcrossingsBefore a b f N ω
· subst hi'
simp only [stoppedValue]
rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)]
by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N
· rw [heq, sub_self]
· rw [sub_nonneg]
exact le_trans (stoppedValue_lowerCrossingTime heq) hf
· rw [sub_eq_zero_of_upcrossingsBefore_lt hab]
rw [Finset.mem_range, not_lt] at hi
exact lt_of_le_of_ne hi (Ne.symm hi')
refine' le_trans _ h₂
rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm]
#align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] :=
calc
(b - a) * μ[upcrossingsBefore a b f N] ≤
μ[∑ k in Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by
rw [← integral_mul_left]
refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _
· exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _)
· refine' eventually_of_forall fun ω => _
simpa using mul_upcrossingsBefore_le (hfN ω) hab
_ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le
_ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero)
#align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
simp_all only [Set.mem_Ici, not_true_eq_false]
· refine' False.elim (h₁ _)
|
simp_all only [Set.mem_Ici]
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine' ⟨fun h => _, fun h => _⟩
· rwa [← sub_le_sub_iff_right a, ←
LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)]
have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by
intro ω i
rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos]
induction' n with k ih
· refine' ⟨rfl, _⟩
simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting,
Set.mem_Icc, Set.mem_Iic, Nat.zero_eq]
ext ω
split_ifs with h₁ h₂ h₂
· simp_rw [hf']
· simp_rw [Set.mem_Iic, ← hf' _ _] at h₂
exact False.elim (h₂ h₁)
· simp_rw [Set.mem_Iic, hf' _ _] at h₁
exact False.elim (h₁ h₂)
· rfl
· have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) =
upperCrossingTime a b f N (k + 1) := by
ext ω
simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right]
split_ifs with h₁ h₂ h₂
· simp_rw [← sub_le_iff_le_add, hf ω]
· refine' False.elim (h₂ _)
simp_all only [Set.mem_Ici, not_true_eq_false]
· refine' False.elim (h₁ _)
|
Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Mathlib_Probability_Martingale_Upcrossing
|
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