A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes
David Landriault Bin Li Hongzhong Zhang





A Unified Approach for Drawdown (Drawup) of
Time-Homogeneous Markov Processes
    David LandriaultDepartment of Statistics and Actuarial Science,
University of Waterloo, Waterloo, ON, N2L 3G1, Canada (dlandria@uwaterloo.ca)
Bin LiCorresponding Author: Department of Statistics and Actuarial
Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
(bin.li@uwaterloo.ca)
Hongzhong ZhangDepartment of IEOR, Columbia University, New York, NY,
10027, USA (hz2244@columbia.edu)
    December 30, 2023
================================================================================================================================================================================================================================================================================================================================================================================================================




Drawdown (resp. drawup) of a stochastic process, also referred as the
reflected process at its supremum (resp. infimum), has wide applications in
many areas including financial risk management, actuarial mathematics and
statistics. In this paper, for general time-homogeneous Markov processes, we
study the joint law of the first passage time of the drawdown (resp. drawup)
process, its overshoot, and the maximum of the underlying process at this
first passage time. By using short-time pathwise analysis, under some mild
regularity conditions, the joint law of the three drawdown quantities is shown
to be the unique solution to an integral equation which is expressed in terms
of fundamental two-sided exit quantities of the underlying process. Explicit
forms for this joint law are found when the Markov process has only one-sided
jumps or is a Lévy process (possibly with two-sided jumps). The proposed
methodology provides a unified approach to
study various drawdown quantities for the general
class of time-homogeneous Markov processes.

Keywords: Drawdown; Integral equation; Reflected process;
Time-homogeneous Markov process

MSC(2000): Primary 60G07; Secondary 60G40



15.5pt



§ INTRODUCTION


We consider a time-homogeneous, real-valued, non-explosive, càdlàg
Markov process X=(X_t)_t≥0 with state space ℝ
[The state space can sometimes be relaxed to
an open interval of ℝ
(e.g., (0,+∞) for geometric Brownian motions).
It is also possible to treat some general state space with complex boundary
behaviors. However, for simplicity, we choose ℝ as the state space of X in this paper.] defined on a filtered probability
space (Ω,ℱ,F=(ℱ_t)_t≥
0,ℙ)
with a complete and right-continuous filtration.
Throughout, we silently assume that X satisfies the strong Markov property (see Section III.8,9 of Rogers and Williams <cit.>), and exclude Markov processes with monotone paths.
The first passage time of X above (below) a level x∈


ℝ is denoted by

    T_x^+(-)=inf{  t≥0:X_t>(<)x},

with the common convention that inf∅=∞.






The drawdown process of X (also known as the reflected process of X at its
supremum) is denoted by Y=(Y_t)_t≥0 with Y_t=M_t-X_t, where
M_t=sup_0≤ s≤ tX_t. Let τ_a=inf{t>0:Y_t>a} be the
first time the magnitude of drawdowns exceeds a given threshold a>0. Note
that (  sup_0≤ s≤ tY_s
>a)  =(  τ_a≤ t) ℙ-a.s. Hence, the
distributional study of the maximum drawdown of X is equivalent to the study
of the stopping time τ_a. Similarly, the drawup process of X is
defined as Ŷ_t=X_t-m_t for t≥0, where m_t=inf_0≤
s≤ tX_t. However, given that the drawup of X can be investigated via
the drawdown of -X, we exclusively focus on the drawdown process Y in this paper.

Applications of drawdowns can be found in many areas. For instance, drawdowns
are widely used by mutual funds and commodity trading advisers to quantify
downside risks. Interested readers are referred to Schuhmacher and Eling
<cit.> for a review of drawdown-based performance measures. An extensive
body of literature exists on the assessment and mitigation of drawdown risks
(e.g., Grossman and Zhou <cit.>, Carr et al. <cit.>, Cherny and
Obloj <cit.>, and Zhang et al. <cit.>). Drawdowns are also closely
related to many problems in mathematical finance, actuarial science and
statistics such as the pricing of Russian options (e.g., Shepp and Shiryaev
<cit.>, Asmussen et al. <cit.> and Avram et al. <cit.>),
De Finetti's dividend problem (e.g., Avram et al.
<cit.> and Loeffen <cit.>), loss-carry-forward taxation models
(e.g., Kyprianou and Zhou <cit.> and Li et al. <cit.>), and
change-point detection methods (e.g., Poor and Hadjiliadis <cit.>). More specifically, in De Finetti's dividend problem under a fixed dividend barrier a>0, the underlying surplus process with dividend payments is a process obtained from reflecting X at a fixed barrier a (the reflected process' dynamics may be different than the drawdown process Y when the underlying process X is not spatial homogeneous). However, the distributional study of ruin quantities in De Finetti's dividend problem can be transformed to the study of drawdown quantities for the underlying surplus process; see Kyprianou and Palmowski <cit.> for a more detailed discussion. Similarly, ruin problems in loss-carry-forward taxation models can also be transformed to a generalized drawdown problem for classical models without taxation, where the generalized drawdown process is defined in the form of Y_t=γ(M_t)-X_t for some measurable function γ(·).

The distributional study of drawdown quantities is not only of theoretical
interest, but also plays a fundamental role in the aforementioned
applications. Early distributional studies on drawdowns date back to Taylor
<cit.> on the joint Laplace transform of τ_a and M_τ_a for
Brownian motions. This result was later generalized by Lehoczky <cit.> to
time-homogeneous diffusion processes. Douady et al. <cit.> and Magdon et
al. <cit.> derived infinite series expansions for the distribution of
τ_a for a standard Brownian motion and a drifted Brownian motion,
respectively. For spectrally negative Lévy processes, Mijatovic and
Pistorius <cit.> obtained a sextuple formula for the joint Laplace
transform of τ_a and the last reset time of the maximum prior to
τ_a, together with the joint distribution of the running maximum, the
running minimum, and the overshoot of Y at τ_a. For some studies on
the joint law of drawdown and drawup of spectrally negative Lévy processes
or diffusion processes, please refer to Pistorius <cit.>, Pospisil et al.
<cit.>, Zhang and Hadjiliadis <cit.>, and Zhang <cit.>.

As mentioned above, Lévy processes[Most often, one-sided Lévy
processes (an exception to this is Baurdoux <cit.> for general Lévy
processes)] and time-homogeneous diffusion processes are two main classes of
Markov processes for which various drawdown problems have been extensively
studied. The treatment of these two classes of Markov processes has typically
been considered distinctly in the literature. For Lévy processes,
Itô's excursion theory is a powerful approach to handle drawdown problems
(e.g., Avram et al. <cit.>, Pistorius <cit.>, and Mijatovic and
Pistorius <cit.>). However, the excursion-theoretic approach is somewhat
specific to the underlying model, and additional care is required when a more
general class of Markov processes is considered. On the other hand, for
time-homogeneous diffusion processes, Lehoczky <cit.> introduced an
ingenious approach which has recently been generalized by many researchers
(e.g., Zhou <cit.>, Li et al. <cit.>, and Zhang <cit.>). Here
again, Lehoczky's approach relies on the continuity of the sample path of the
underlying model, and hence is not applicable for processes with upward jumps.
Also, other general methodologies (such as the martingale approach in, e.g.,
Asmussen <cit.> and the occupation density approach in, e.g., Ivanovs
and Palmowski <cit.>) are well documented in the literature but they
strongly depend on the specific structure of the underlying process. To the
best of our knowledge, no unified treatment of drawdowns (drawups) for general
Markov processes has been proposed in the literature.

In this paper, we propose a general and unified approach to study the joint
law of (τ_a,M_τ_a,Y_τ_a) for time-homogeneous Markov
processes with possibly two-sided jumps. Under mild regularity conditions, the
joint law is expressed as the solution to an integral equation which involves
two-sided exit quantities of the underlying process X. The uniqueness of the
integral equation for the joint law is also investigated. In particular, the
joint law possesses explicit forms when X has only one-sided jumps or is a
Lévy process (possibly with two-sided jumps). In general, our main result
reduces the drawdown problem to fundamental two-sided exit quantities.

The main idea of our proposed approach is briefly summarized below. By
analyzing the evolution of sample paths over a short time period following
time 0 and using renewal arguments, we first establish tight upper and lower
bounds for the joint law of (τ_a,M_τ_a,Y_τ_a) in terms of
the two-sided exit quantities. Then, under mild regularity conditions, we use
a Fatou's lemma with varying measures to show that the upper and lower bounds
converge when the length of the time interval approaches 0. This leads to an
integro-differential equation satisfied by the desired joint law. Finally, we
reduce the integro-differential equation to an integral equation. When X is
a spectrally negative Markov process or a general Lévy process, the
integral equation can be solved and the joint law of (τ_a,M_τ_a
,Y_τ_a) is hence explicitly expressed in terms of two-sided exit quantities.

The rest of the paper is organized as follows. In Section 2, we introduce some
fundamental two-sided exit quantities and present several preliminary results.
In Section 3, we derive the joint law of (τ_a,Y_τ_a,M_τ_a
) for general time-homogeneous Markov processes. Several Markov processes
for which the proposed regularity conditions are met are further discussed.
Some numerical examples are investigated in more detail in Section 4.
Some technical proofs are postponed to Appendix.



§ PRELIMINARY


For ease of notation, we adopt the following conventions throughout the paper.
We denote by ℙ_x the law of X given X_0=x∈


ℝ and write ℙ≡ℙ_0 for brevity. We write u∧
v=min{u,v}, ℝ
_+=[0,∞), and ∫_x^y·dz for an integral on the
open interval z∈(x,y).

For q,s≥0, u≤ x≤ v and z>0, we introduce the following
two-sided exit quantities of X:

    B_1^(q)(x;u,v)       :=𝔼_x[  e^-qT_v^+1_{T_v^+<∞,  
    T_v^+<T_u^-,X_T_v^+=v}]  ,
    
    B_2^(q)(x,dz;u,v)       :=𝔼_x[  e^-qT_v^+
    1_{T_v^+<∞,    T_v^+<T_u^-,X_T_v^+-v∈dz}]  ,
    
    C^(q,s)(x;u,v)       :=𝔼_x[  e^-qT_u^--s(u-X_T_u^-
    )1_{ T_u^-<∞,   T_u^-<T_v^+}]  .

We also define the joint Laplace transform

    B^(q,s)(x;u,v):=𝔼_x[  e^-qT_v^+-s(X_T_v^+
    -v)1_{ T_v^+<∞,  T_v^+<T_u^-}]  =B_1^(q)
    (x;u,v)+B_2^(q,s)(x;u,v),

where B_2^(q,s)(x;u,v):=∫_0^∞e^-szB_2^(q)
(x,dz;u,v).

The following pathwise inequalities are central to the construction of tight
bounds for the joint law of the triplet (τ_a,M_τ_a,Y_τ_a).


For q,s≥0, x∈ℝ and ε∈(0,a),
we have ℙ_x-a.s.

    1_{T_x+ε^+<∞, T_x+ε^+<T_x+ε-a^-}≤1_{T_x+ε^+<∞, T_x+ε^+<τ_a}≤1_{T_x+ε^+<∞, T_x+ε^+<T_x-a^-},

and

    e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞,  τ_a<T_x+ε
    ^+}   ≥ e^-qT_x-a^--s(x-a-X_T_x-a^-)-sε1_{T_x-a^-<∞, T_x-a^-<T_x+ε^+},
    
    e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞, τ_a<T_x+ε
    ^+}   ≤ e^-qT_x+ε-a^-
    -s(x-a-X_T_x+ε-a^-)1_{T_x+ε-a^-<∞, T_x+ε-a^-
    <T_x+ε^+}.





By analyzing the sample paths of X, it is easy to see that, for any path ω∈(T_x+ε^+<∞), we have _x{τ_a≤ T_x-a^-}=1, so


    (T_x+ε^+<∞, T_x+ε^+<τ_a)=(T_x+ε^+<∞, T_x+ε^+<τ_a≤ T_x-a^-)⊂(T_x+ε^+<∞,  T_x+ε^+<T_x-a^-)  ℙ_x-a.s.

and similarly, _x-a.s.

    (T_x+ε^+<∞, T_x+ε^+<T_x+ε-a^-)=(T_x+ε^+<∞, T_x+ε^+< T_x+ε-a^-, T_x+ε^+< τ_a)⊂(T_x+ε^+<∞,  T_x+ε^+<τ_a),

which immediately implies (<ref>). On the other hand, by using the same argument, we have

    (T_x-a^-<∞, T_x-a^-<T_x+ε^+)=(T_x-a^-<∞, τ_a≤ T_x-a^-<T_x+ε^+)⊂(τ_a<∞, τ_a<T_x+ε^+)  ℙ_x-a.s.

and

    (τ_a<∞, τ_a<T_x+ε^+)=(τ_a<∞, T_x+ε-a^-≤τ
    _a<T_x+ε^+)⊂(T_x+ε-a^-<∞, T_x+ε-a^-<T_x+ε^+)  ℙ_x-a.s.

For any path ω∈(T_x-a^-<∞, T_x-a^-<T_x+ε^+), we know
from (<ref>) that ω∈(T_x-a^-<∞, τ_a≤ T_x-a^-<T_x+ε
^+). This implies M_τ_a(ω)≤ x+ε and X_τ
_a(ω)≥ X_T_x-a^-(ω), which further entails that
Y_τ_a(ω)=M_τ_a(ω)-X_τ_a(ω)≤
x+ε-X_T_x-a^-(ω). Therefore, by the above analysis and
the second inequality of (<ref>),

    e^-qT_x-a^--s(x+ε-X_T_x-a^-)1_{ T_x-a^-<∞, T_x-a
    ^-<T_x+ε^+}≤ e^-qτ_a-sY_τ_a
    1_{τ_a<∞,  τ_a<T_x+ε^+}  ℙ
    _x-a.s.

which naturally leads to (<ref>).

Similarly, for any sample path ω∈(τ_a<∞, τ_a<T_x+ε^+),
we know from (<ref>) that ω∈(τ_a<∞, T_x+ε-a^-≤τ
_a<T_x+ε^+), which implies that x-X_T_x+ε
-a^-(ω)≤ Y_T_x+ε-a^-(ω)≤ Y_τ_a
(ω). Therefore, by the first inequality of (<ref>), we obtain

    e^-qτ_a-sY_τ_a1_{τ_a<∞, τ_a<T_x+ε
    ^+}≤ e^-qT_x+ε-a^--s(x-X_T_x+ε
    -a^-)1_{T_x+ε-a^-<∞, T_x+ε-a^-<T_x+ε^+}  ℙ_x-a.s.

This implies the second inequality of (<ref>).

By Proposition <ref>, we easily obtain the following useful estimates.


For q,s≥0, x∈


ℝ
,z>0 and ε∈(0,a),

    B_1^(q)(x;x+ε-a,x+ε)     ≤𝔼_x[
    e^-qT_x+ε^+1_{T_x+ε^+<∞, T_x+ε^+<τ_a
    ,X_T_x+ε^+=x+ε}]  ≤ B_1^(q)
    (x;x-a,x+ε),
    
    B_2^(q)(x,dz;x+ε-a,x+ε)     ≤𝔼_x[  e^-qT_x+ε^+1_{T_x+ε^+<∞, T_x+ε
    ^+<τ_a,X_T_x+ε^+-x-ε∈dz}]
    ≤ B_2^(q)(x,dz;x-a,x+ε),

and

    e^-sεC^(q,s)(x;x-a,x+ε)≤𝔼_x[
    e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞,  τ_a<T_x+ε
    ^+}]  ≤ e^sεC^(q,s)(x;x+ε
    -a,x+ε).





It is not difficult to check that the results of Proposition
<ref> and Corollary <ref> still hold if the first passage
times and the drawdown times are only observed discretely or randomly (such as
the Poisson observation framework in Albrecher et al. <cit.> for the
latter). Further, explicit relationship between Poisson observed first passage
times and Poisson observed drawdown times (similar as for Theorem
<ref> below) can be found by exploiting the same approach as laid
out in this paper.


The later analysis involves the weak convergence of measures which is recalled
here. Consider a metric space S with the Borel σ-algebra on it. We
say a sequence of finite measures {μ_n}_n∈


ℕ is weakly convergent to a finite measure μ as n→∞ if

    lim_n→∞∫_Sϕ(z)dμ_n(z)=∫_S
    ϕ(z)dμ(z),

for any bounded and continuous function ϕ(·) on S.

In the next lemma, we show some forms of Fatou's lemma for varying measures
under weak convergence. Similar results are proved in Feinberg et al.
<cit.> for probability measures. For completeness, a proof for general
finite measures is provided in Appendix.


Suppose that {μ_n}_n∈


ℕ is a sequence of finite measures on S which is weakly convergent to a
finite measure μ, and {ϕ_n}_n∈


ℕ is a sequence of uniformly bounded and nonnegative functions on S. Then,

    ∫_Slim inf_n→∞,w→ zϕ_n(w)d
    μ(z)≤lim inf_n→∞∫_Sϕ_n(z)dμ
    _n(z),

and

    ∫_Slim sup_n→∞,w→ zϕ_n(w)d
    μ(z)≥lim sup_n→∞∫_Sϕ_n(z)dμ_n(z).






§ MAIN RESULTS


In this section, we study the joint law of (τ_a,M_τ_a,Y_τ
_a) for a general Markov process with possibly two-sided jumps. The
following assumptions on the two-sided exit quantities of X are assumed to
hold, which are sufficient (but not necessary) conditions for the
applicability of our proposed methodology. Weaker assumptions might be assumed
for special Markov processes; see, for instance, Remark <ref> and
Corollary <ref> below.


For all q,s≥0, z>0 and x>X_0, we assume the following limits exist
and identities hold:

    (A1)b_a,1^(q)(x)       :=lim_ε↓0
    1-B_1^(q)(x;x-a,x+ε)/ε=lim_ε↓01-B_1^(q)(x;x+ε-a,x+ε)/ε
         =lim_ε↓01-B_1^(q)(x-ε
    ;x-a,x)/ε=lim_ε↓01-B_1
    ^(q)(x-ε;x-ε-a,x)/ε,

and ∫_x^yb_a,1^(q)(w)dw<∞ for any x,y∈


ℝ;

    (A2)b_a,2^(q,s)(x)       :=lim_ε↓
    01/εB_2^(q,s)(x;x-a,x+ε)=lim_ε↓01/εB_2^(q,s)(x;x+ε-a,x+ε
    )
         =lim_ε↓01/εB_2^(q,s)
    (x-ε;x-a,x)=lim_ε↓01/ε
    B_2^(q,s)(x-ε;x-ε-a,x),

and s⟼ b_a,2^(q,s)(x) is right continuous at s=0;

    (A3)c_a^(q,s)(x)       :=lim_ε↓0
    C^(q,s)(x;x-a,x+ε)/ε=lim_ε↓0C^(q,s)(x;x+ε-a,x+ε)/ε
         =lim_ε↓0C^(q,s)(x-ε;x-a,x)/ε=lim_ε↓0C^(q,s)(x-ε
    ;x-ε-a,x)/ε.




Under Assumptions (A1) and (A2), it follows from (<ref>)
that

    b_a^(q,s)(x):=lim_ε↓01-B^(q,s)
    (x;x-a,x+ε)/ε=b_a,1^(q)(x)-b_a,2^(q,s)(x).




 Due to the general structure of X, it is difficult
to refine Assumptions (A1)-(A3) unless a specific structure
for X is given. A necessary condition for
Assumptions (A1)-(A3) to hold is that,

    T_x^+=0 and X_T_x^+=x,ℙ_x-a.s. for
    all x∈ℝ.

In other words, X must be upward regular and creeping upward at every
x.[See page 142 and page 197 of <cit.> for definitions of
regularity and creeping for Lévy processes.] In the later part of this
section, we provide some examples of Markov processes which satisfy
Assumptions (A1)-(A3), including spectrally negative
Lévy processes, linear diffusions, piecewise exponential Markov processes,
and jump diffusions.



By Theorem 5.22 of Kallenberg <cit.> or
Proposition 7.1 of Landriault et al. <cit.>, we know that Assumption
(A2) implies that the measures 1/εB_2
^(q)(x,dz;x-a,x+ε), 1/εB_2
^(q)(x,dz;x+ε-a,x+ε), 1/εB_2^(q)(x-ε,dz;x-a,x) and 1/ε
B_2^(q)(x-ε,dz;x-ε-a,x) weakly converge to
the same measure on ℝ
_+, denoted as b_a,2^(q)(x,dz), such that ∫_


ℝ
_+e^-szb_a,2^(q)(x,dz)=b_a,2^(q,s)(x). We point out
that it is possible that b_a,2^(q)(x,{0})>0, though the measure
B_2^(q)(x,dz;u,v) is only defined on z∈(0,∞).


We are now ready to present the main result of this paper related to the joint
law of (τ_a,Y_τ_a,M_τ_a).


Consider a general time-homogeneous Markov process X
satisfying Assumptions (A1)-(A3). For q,s≥0 and
K∈ℝ, let

    h(x)=𝔼_x[  e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞, M_τ_a
    ≤ K}]  ,   x≤ K.

Then h(·) is differentiable in x<K and solves the following integral
equation

    h(x)=∫_x^Ke^-∫_x^yb_a,1^(q)(w)dw(
    c_a^(q,s)(y)+∫_[0,K-y)h(y+z)b_a,2^(q)(y,dz))
    dy,   x≤ K.





By the strong Markov property of X, for any X_0=x≤ y<K and
0<ε<(K-y)∧ a, we have

    h(y)       =𝔼_y[  e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞, 
    τ_a<T_y+ε^+}]  +𝔼_y[
    e^-qT_y+ε^+1_{T_y+ε^+<∞, T_y+ε^+<τ_a
    ,X_T_y+ε^+=y+ε}]  h(y+ε)
         +∫_0^K-y-ε𝔼_y[  e^-qT_y+ε
    ^+1_{T_y+ε^+<∞, T_y+ε^+<τ_a,X_T_y+ε^+
    -y-ε∈dz}]  h(y+ε+z).

By Corollary <ref>, it follows that

    h(y+ε)-h(y)     ≥-e^sεC^(q,s)(y;y+ε
    -a,y+ε)+(  1-B_1^(q)(y;y-a,y+ε))
    h(y+ε)
         -∫_0^K-y-εh(y+ε+z)B_2^(q)(y,d
    z;y-a,y+ε),

and

    h(y+ε)-h(y)     ≤-e^-sεC^(q,s)(y;y-a,y+ε
    )+(  1-B_1^(q)(y;y+ε-a,y+ε))
    h(y+ε)
         -∫_0^K-y-εh(y+ε+z)B_2^(q)(y,d
    z;y+ε-a,y+ε).

By Assumptions (A1)-(A3) and h(·)∈0,1], it is
clear that both the lower bound of h(y+ε)-h(y) in (<ref>) and
the upper bound in (<ref>) vanish as ε↓0. Hence,
h(y) is right continuous for y∈ x,K). Replacing y by
y-ε in (<ref>) and (<ref>), and using Assumptions
(A1)-(A3) again, it follows that h(y) is also left
continuous for y∈(x,K] with h(K)=0. Therefore, h(y) is continuous for
y∈ x,K] (left continuous at x and right continuous at K).

To consecutively show the differentiability, we divide inequalities
(<ref>) and (<ref>) by ε. It follows from Assumptions
(A1)-(A3), Remark <ref>, Lemma <ref> and
the continuity of h that

    lim inf_ε↓0h(y+ε)-h(y)/ε
       ≥-c_a^(q,s)(y)+b_a,1^(q)(y)h(y)-lim sup_ε↓
    0∫_0^K-y-εh(y+ε+z)B_2^(q)(y,d
    z;y-a,y+ε)/ε
       ≥-c_a^(q,s)(y)+b_a,1^(q)(y)h(y)-∫_[0,K-y)h(y+z)b_a,2
    ^(q)(y,dz),

and similarly,

    lim sup_ε↓0h(y+ε)-h(y)/ε
    ≤-c_a^(q,s)(y)+b_a,1^(q)(y)h(y)-∫_[0,K-y)h(y+z)b_a,2
    ^(q)(y,dz).

Since the two limits coincide, one concludes that h(y) is right
differentiable for y∈(x,K). Moreover, by replacing y by y-ε
in (<ref>) and (<ref>), and using similar arguments, we can show that
h(y) is also left differentiable for y∈(x,K). Since the left and right
derivatives coincide, we conclude that h(y) is differentiable for any
y∈(x,K) and solves the following ordinary integro-differential equation
(OIDE),

    h^'(y)-b_a,1^(q)(y)h(y)=-c_a^(q,s)(y)-∫_[0,K-y)
    h(y+z)b_a,2^(q)(y,dz).



Multiplying both sides of (<ref>) by e^-∫_x^yb_a,1
^(q)(w)dw, integrating the resulting equation (with respect to
y) from x to K, and using h(K)=0, this completes the proof of Theorem
<ref>.

When the Markov process X is spectrally negative (i.e., with no upward
jumps), the upward overshooting density b_a,2^(q)(x,dz) is
trivially 0. Theorem <ref> reduces to the following corollary.


Consider a spectrally negative time-homogeneous Markov process
X satisfying Assumptions (A1) and (A3). For q,s≥0 and
K>0, we have

    𝔼_x[  e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞, M_τ_a≤
    K}]  =∫_x^Ke^-∫_x^yb_a,1^(q)(w)dw
    c_a^(q,s)(y)dy,   x≤ K.




When X is a general Lévy process (possibly with two-sided jumps), we
have the following result for the joint Laplace transform of the triplet
(τ_a,Y_τ_a,M_τ_a). Note that Corollary <ref>
should be compared to Theorem 4.1 of Baurdoux <cit.>, in which, under the
Lévy framework, the resolvent density of Y is expressed in terms of the
resolvent density of X using excursion theory.


Consider a Lévy process X satisfying Assumptions
(A1)-(A3). For q,s,δ≥0, we have[For Lévy processes {τ_a<∞}=1 as long as X is not monotone.]

    𝔼[  e^-qτ_a-s(Y_τ_a-a)-δ M_τ_a]
    =c_a^(q,s)(0)/δ+b_a^(q,δ)(0).





By the spatial homogeneity of the Lévy process X, Eq. (<ref>)
at x=0 reduces to

    h(0)=c_a^(q,s)(0)/b_a,1^(q)(0)(  1-e^-b_a,1^(q)
    (0)K)  +∫_0^Ke^-b_a,1^(q)(0)y∫_[0,K-y)h(y+z)b_a,2
    ^(q)(0,dz)dy.

Let

    ĥ(0):=𝔼[  e^-qτ_a-s(Y_τ_a-a)-δ
    M_τ_a]  =𝔼[  e^-qτ_a-s(Y_τ_a
    -a)1_{M_τ_a≤ e_δ}]  ,

where e_δ is an independent exponential random variable with finite
mean 1/δ>0. Multiplying both sides of (<ref>) by δ e^-δ
K, integrating the resulting equation (with respect to K) from 0 to
∞, and using integration by parts, one obtains

    ĥ(0)       =c_a^(q,s)(0)/δ+b_a,1^(q)(0)+∫
    _0^∞δ e^-δ K∫_0^Ke^-b_a,1^(q)(0)y
    ∫_[0,K-y)h(y+z)b_a,2^(q)(0,dz)dydK
         =c_a^(q,s)(0)/δ+b_a,1^(q)(0)+∫_0^∞e^-b_a,1^(q)(0)ydy∫_
    
    
    ℝ
    _+b_a,2^(q)(0,dz)∫_z+y^∞δ e^-δ
    K𝔼[  e^-qτ_a-s(Y_τ_a-a)1_{M_τ_a≤
    K-y-z}]  dK
         =c_a^(q,s)(0)/δ+b_a,1^(q)(0)+ĥ(0)∫_
    
    
    ℝ
    _+e^-δ zb_a,2^(q)(0,dz)/δ+b_a,1^(q)(0).

Solving for ĥ(0) and using (<ref>), it follows that

    ĥ(0)=c_a^(q,s)(0)/δ+b_a,1^(q)(0)-∫_
    
    
    ℝ
    _+e^-δ zb_a,2^(q)(0,dz)=c_a^(q,s)(0)/δ+b_a^(q,δ)(0).

It follows from the monotone convergence theorem that (<ref>) also holds
for δ=0.


 We point out that Assumptions (A1)-(A3) are not necessary to
yield (<ref>) in the Lévy framework. In fact, by the spatial
homogeneity of X, similar to (<ref>) and (<ref>), we have

    e^-(s+δ)εC^(q,s)(0;-a,ε)/1-e^-δεB^(q,δ)(0;ε-a,ε)≤𝔼[
    e^-qτ_a-s(Y_τ_a-a)-δ M_τ_a]  ≤e^sεC^(q,s)(0;ε-a,ε)/1-e^-δεB^(q,δ)(0;-a,ε),

for any ε∈(0,a). Suppose that the following condition holds:

    lim_ε↓0C^(q,s)(0;-a,ε)/1-e^-δεB^(q,δ)(0;ε-a,ε)=lim_ε↓0C^(q,s)(0;ε-a,ε)/1-e^-δεB^(q,δ)(0;-a,ε):=D_a^(q,s,δ)

Then,

    𝔼[  e^-qτ_a-s(Y_τ_a-a)-δ M_τ_a]
    =D_a^(q,s,δ).




Theorem <ref> shows that the joint law 𝔼_x[
e^-qτ_a-s(Y_τ_a-a)1_{M_τ_a≤ K}] is a
solution to Eq. (<ref>). Furthermore, the following theorem shows
that Eq. (<ref>) admits a unique solution.


Suppose that Assumptions (A1)-(A3) hold. For q,s≥0 and
K>0, Eq. (<ref>) admits a unique solution.



From Theorem <ref>, we know that h(x):=𝔼_x[
e^-qτ_a-s(Y_τ_a-a)1_{τ_a<∞, M_τ_a≤ K}] is a
solution of (<ref>). We also notice that any continuous solution to
(<ref>) must vanish when x↑ K. For any fixed
L∈(-∞,K), we define a metric space (𝔸_L,d
_L), where 𝔸_L={  f∈ C[L,K],f(K)=0} and the
metric d_L(f,g)=sup_x∈ L,K]|f(x)-g(x)| for
f,g∈𝔸_L. We then define a mapping ℒ on
𝔸_L by

    ℒf(x)=∫_x^Ke^-∫_x^yb_a,1^(q)(w)d
    w(  c_a^(q,s)(y)+∫_[0,K-y)f(y+z)b_a,2^(q)(y,d
    z))  dy,  x∈ L,K],

where f∈𝔸_L. It is clear that ℒ(𝔸
_L)⊂𝔸_L.

Next we show that ℒ:𝔸_L→𝔸_L is a
contraction mapping. By the definitions of the two-sided exit quantities, for
any y∈


ℝ, it follows that

    C^(q,s)(y;y-a,y+ε)+∫_
    
    
    ℝ
    _+B_2^(q)(y,dz;y-a,y+ε)≤1-B_1^(q)
    (y;y-a,y+ε).

Dividing each term in (<ref>) by ε∈(0,a) and letting
ε↓0, it follows from Assumptions (A1
)-(A3) that

    0≤ c_a^(q,s)(y)+∫_
    
    
    ℝ
    _+b_a,2^(q)(y,dz)≤ b_a,1^(q)(y),   y∈
    
    
    ℝ
    .

By (<ref>), we have for any f,g∈𝔸_L,

    d_L(  ℒf,ℒg)      ≤sup
    _t∈ L,K]| f(t)-g(t)|sup_x∈ L,K]
    ∫_x^Ke^-∫_x^yb_a,1^(q)(w)dw∫_
    
    
    ℝ
    _+b_a,2^(q)(y,dz)dy
       ≤d_L(f,g)sup_L≤ x≤ K∫_x^Ke^-∫_x
    ^yb_a,1^(q)(w)dwb_a,1^(q)(y)dy
       ≤d_L(f,g)(  1-e^-∫_L^Kb_a,1^(q)
    (w)dw)  .

Since ∫_L^Kb_a,1^(q)(w)dw<∞ by Assumption
(A1), one concludes that ℒ:𝔸_L→𝔸_L is a contraction mapping. By Banach fixed point theorem, there
exists a unique fixed point in 𝔸_L. By a restriction of domain,
it is easy to see that 𝔸_L_1⊂𝔸_L_2 for
-∞<L_1<L_2<K. By the arbitrariness of L, the uniqueness holds for
the space ∩_L<K𝔸_L. This completes the proof.


For the reminder of this section, we state several examples of Markov
processes satisfying Assumptions (A1)-(A3). Note that the
joint law of drawdown estimates for Examples <ref> and
<ref> were solved by Mijatovic and Pistorius <cit.> and
Lehoczky <cit.>, respectively (using different approaches). Assumption
verifications for Examples <ref> and <ref> are postponed to Appendix.


[Spectrally negative Lévy processes] Consider a
spectrally negative Lévy process X. Let ψ(s):=1/t
log𝔼[e^sX_t] (  s≥0) be the Laplace exponent
of X. Further, let W^(q):


ℝ
→0,∞) be the well-known q-scale function of X; see,
for instance Chapter 8 of Kyprianou <cit.>. The second scale function is
defined as Z^(q)(x)=1+q∫_0^xW^(q)(y)dy. Under some mild
conditions (e.g., Lemma 2.4 of Kuznetsov et al. <cit.>), the scale
functions are continuously differentiable which further implies that
Assumptions (A1) and (A3) hold with

    b_a,1^(q)(0)=W^(q)'(a)/W^(q)(a) and c_a
    ^(q,s)(0)=e^saZ_s^(p)(a)W_s^(p)'(a)-Z_s^(p)'(a)W_s^(p)(a)/W_s^(p)(a),

where p=q-ψ(s), and W_s^(p) (Z_s^(p)) is the (second) scale
function of X under a new probability measure ℙ^s defined by
the Radon-Nikodym derivative process .  dℙ^s
/dℙ| _ℱ_t=e^sX_t-ψ(s)t for
t≥0. Therefore, by Corollary <ref> and (<ref>), we have

    𝔼[  e^-qτ_a-s(Y_τ_a-a)-δ M_τ_a]
    =e^saW^(q)(a)/δ W^(q)(a)+W^(q)'(a)Z_s
    ^(p)(a)W_s^(p)'(a)-pW_s^(p)(a)^2/W_s^(p)(a),

which is consistent with Theorem 3.1 of Landriault et al. <cit.>, and
Theorem 1 of
Mijatovic and Pistorius <cit.>.



[Refracted Lévy processes] Consider a
refracted spectrally negative Lévy process X of the form

    X_t=U_t-λ∫_0^t1_{X_s>b}ds,

where λ≥0, b>0, and U is a spectrally negative Lévy process
(see Kyprianou and Loeffen <cit.>). Let W^(q) (Z^(q)) be the
(second) q-scale function of U, and 𝕎^(q) be the q-scale
function of the process {U_t-λ t}_t≥0. Similar to Example
<ref>, all the scale functions are continuously differentiable under
mild conditions.

For simplicity, we only consider the quantity
𝔼_x[
e^-qτ_a1_{τ_a<∞, M_τ_a≤ K}] with b>x-a (otherwise the
problem reduces to Example <ref> for X_t=U_t-λ t). By
Theorem 4 of Kyprianou and Loeffen <cit.>, one can verify that
Assumptions (A1) and (A3) hold. For b>x, from (<ref>)
with s=0, we have

    b_a,1^(q)(x)=W^(q)'(a)/W^(q)(a) and c_a
    ^(q,0)(x)=Z^(q)(a)W^(q)'(a)-Z^(q)'(a)W^(q)
    (a)/W^(q)(a).

For x>b>x-a,

    b_a,1^(q)(x)=(  1+λ𝕎^(q)(0))
    W^(q)'(a)+λ∫_b-x+a^a𝕎^(q)'
    (a-y)W^(q)'(y)dy/W^(q)(a)+λ∫_b-x+a^a
    𝕎^(q)(a-y)W^(q)'(y)dy

and

    c_a^(q,0)(x)=k_a^(q)(x)/W^(q)(a)+λ∫_b-x+a
    ^a𝕎^(q)(a-y)W^(q)'(y)dy,

where

    k_a^(q)(x)       =(1+λ𝕎^(q)(0))(  Z^(q)
    (a)W^(q)'(a)-qW^(q)(a)^2) 
         +λ q(1+λ𝕎^(q)(0))∫_b-x+a^a𝕎
    ^(q)(a-y)(  W^(q)'(a)W^(q)(y)-W^(q)(a)W^(q)'(y))  dy
         -λ q[  W^(q)(a)+λ∫_b-x+a^a𝕎
    ^(q)(a-y)W^(q)'(y)dy]  ∫_b-x+a^a𝕎
    ^(q)'(a-y)W^(q)(y)dy
         +λ[  Z^(q)(a)+λ q∫_b-x+a^a𝕎
    ^(q)(a-y)W^(q)(y)dy]  ∫_b-x+a^a𝕎
    ^(q)'(a-y)W^(q)'(y)dy.

















By Corollary <ref>, we obtain

    𝔼_x[  e^-qτ_a1_{M_τ_a≤ K}]
    =∫_x^Ke^-∫_x^yb_a,1^(q)(w)dwc_a^(q,0)
    (y)dy,   x≤ K,

which is a new result for the refracted Lévy process (<ref>).



[Linear diffusion processes] Consider a linear
diffusion process X of the form

    dX_t=μ(X_t)dt+σ(X_t)dW_t,

where (W_t)_t≥0 is a standard Brownian motion, and the drift term
μ(·) and local volatility σ(·)>0 satisfy the usual
Lipschitz continuity and linear growth conditions. As a special case of the
jump diffusion process of Example <ref>, it will be shown later that
Assumptions (A1) and (A3) hold for linear diffusion
processes. By Corollary <ref>, we obtain

    𝔼_x[  e^-qτ_a1_{τ_a<∞, M_τ_a≤ K}]
    =∫_x^Ke^-∫_x^yb_a,1^(q)(w)dwc_a^(q,0)
    (y)dy,   x≤ K,

which is consistent with Eq. (4) of Lehoczky <cit.>.



[Piecewise exponential Markov processes] Consider a
piecewise exponential Markov process (PEMP) X of the form

    dX_t=μ X_tdt+dZ_t,

where μ>0 is the drift coefficient and Z=(Z_t)_t≥0 is a compound
Poisson process given by Z_t=∑_i=1^N_tJ_i. Here, (N_t
)_t≥0 is a Poisson process with intensity λ>0 and J_i's are
iid copies of a real-valued random variable J with cumulative distribution
function F.
We also assume the initial value X_0≥ a which ensures that X_t≥ 0 for all t<τ_a. In this case, as discussed in Remark <ref>, X is upward regular and creeps upward before τ_a.


The first passage times of X have been extensively studied in applied
probability; see, e.g., Tsurui and Osaki <cit.> and Kella and Stadje
<cit.>. For the PEMP (<ref>), semi-explicit expressions for the
two-sided exit quantities B_1^(q)(·), B_2^(q)(·,·) and
C^(q,s)(·) are given in Section 6 of Jacobsen and Jensen <cit.>.
As will be shown in Section <ref>, Assumptions (A1
)-(A3) and Theorem <ref> hold for the PEMP X with a
continuous jump size distribution F.



[Jump diffusion] Consider a jump diffusion process
X of the form

    dX_t=μ(X_t)dt+σ(X_t)dW_t
    +∫_-∞^∞γ(X_t-,z)N(dt,dz),

where μ(·) and σ(·)>0 are functions on ℝ,
(W_t)_t≥0 is a standard Brownian motion, γ(·,·) is a
real-valued function on ℝ^2 modeling the jump size, and
N(dt,dz) is an independent Poisson random measure on
ℝ_+×ℝ with a finite intensity measure
dt×ν(dz). For
specific μ(·) and σ(·), the jump diffusion (<ref>) can
be used to model the surplus process of an insurer with investment in risky
assets; see, e.g., Gjessing and Paulsen <cit.> and Yuen et al.
<cit.>. We assume the same conditions as Theorem 1.19 of Øksendal and
Sulem-Bialobroda <cit.> so that (<ref>) admits a unique càdlàg
adapted solution. Under this setup, we show in Section <ref> that
Assumptions (A1)–(A3) and thus Theorem <ref>
hold for the jump diffusion (<ref>).




§ NUMERICAL EXAMPLES


The main results of Section 3 rely on the analytic tractability of the
two-sided exit quantities. To further illustrate their applicability, we now
consider the numerical evaluation of the joint law of (Y_τ_a
,M_τ_a) for two particular spatial-inhomogeneous Markov processes with
(positive) jumps through Theorem <ref>. For simplicity, we assume
that the discount rate q=0 throughout this section.



 §.§ PEMP


In this section, we consider the PEMP X in Example <ref> with
μ=1, λ=3, and the generic jump size J with density

    p(x)={[        1/3e^-x,            x>0,; 1/3(e^x+2e^2x),            x<0. ].

We follow Section 6 of Jacobsen and Jensen <cit.> to first solve for the
two-sided exit quantities. Define the integral kernel

    ψ_0(z):=1/z(z+1)(z-1)(z-2),   z∈
    
    
    ℂ
    ,

and the linearly independent functions

    [ g_1(x):=1/2π√(-1)∫_Γ_1ψ_0(z)e^-xz
    dz=1/6e^-2x, g_2(x):=1/2π√(-1)∫_Γ_2
    ψ_0(z)e^-xzdz=-1/2e^-x,;      g_3(x):=1/2π√(-1)∫_Γ_3ψ_0(z)e^-xz
    dz=1/2,  g_4(x):=1/2π√(-1)∫_Γ_4ψ
    _0(z)e^-xzdz=-1/6e^x, ]

for x>0, where Γ_i (i=1,2,3,4) is a small counterclockwise circle
centered at the pole μ_i=3-i of ψ_0(z). Moreover, for 0<u<v, we
consider the matrix-valued function

    (M_i,k(u,v))_1≤ i,k≤4:=
    [ -1/3e^-2u(u+11/6)           e^-2u/6          e^-2v/18            g_1(v);              e^-u     e^-u/2(u+1/2)           -e^-v/4            g_2(v);              -1/2              -1/2               1/2            g_3(v);             e^u/9            e^u/12     e^v/6(v-11/6)            g_4(v) ]
    ,

where the matrix M entries are chosen according to

    {[ M_i,k(u,v)=μ_k/2π√(-1)∫_Γ_iψ_0
    (z)/z-μ_ke^-uzdz,  1≤ i≤4, k=1,2,;      M_i,3(u,v)=|μ_4|/2π√(-1)∫_Γ_iψ
    _0(z)/z-μ_4e^-vzdz,  1≤ i≤4. ].

Let (N_k,j(u,v))_1≤ k,j≤4 be the inverse of (M_i,k(u,v))_1≤
i,k≤4.
Combining Eq. (46) and a generalized Eq. (48) of Jacobsen and Jensen
<cit.> (with ζ=s≥0 and ρ≥0), we obtain the linear system
of equations

    (c_1,c_2,c_3,c_4)(M_i,k)=(  -2C/s+2
    ,-C/s+1,C/ρ+1,f(v))  ,

where C and C are constants specified later, and
f(x) could stand for any of B_1^(0)(x;u,v), B_2^(0,ρ)(x;u,v),
or C^(0,s)(x;u,v) and has the representation

    f(x)=∑_i=1^4c_ig_i(x),   x∈ u,v].



To solve for B_1^(0)(x;u,v), B_2^(0,ρ)(x;u,v), or
C^(0,s)(x;u,v), we only need to solve (<ref>) with different assigned
values of C, C, and f(v) according to Eq. (45) of
Jacobsen and Jensen <cit.>. By letting C=C=0 and
f(v)=1, we obtain

    B_1^(0)(x;u,v)=∑_i=1^4N_4,i(u,v)g_i(x).

Similarly, by letting C=f(v)=0 and C=1, for
ρ≥0, we obtain

    B_2^(0,ρ)(x;u,v)=1/1+ρ∑_i=1^4N_3,i(u,v)g_i(x).

A Laplace inversion with respect to ρ yields, for z>0,

    B_2^(0)(x,dz;u,v)=e^-z∑_i=1^4N_3,i(u,v)g_i
    (x)dz.

By letting C=1 and C=f(v)=0, for s≥0, we
obtain

    C^(0,s)(x;u,v)=∑_i=1^4(  -2/s+2N_1,i(u,v)+-1/s+1N_2,i(u,v))  g_i(x).

By the definitions, we have

    b_a,1^(0)(x)       =-∑_i=1^4D_4,i(x-a,x)g_i(x),
    
    b_a,2^(0)(x,dz)       =e^-z(  ∑_i=1^4D_3,i
    (x-a,x)g_i(x))  dz,
    
    c_a^(0,s)(x)       =∑_i=1^4(  -2/s+2D_1,i
    (x-a,x)+-1/s+1D_2,i(x-a,x))  g_i(x),

where we denote D_k,j(u,v):=∂/∂ vN_k,j(u,v).

In Figure <ref> below, we use  to numerically solve
the integral equation (<ref>).





 §.§ A jump diffusion model


In this section, we consider a generalized PEMP (X_t)_t≥0 with
diffusion whose dynamics is governed by

    dX_t=X_tdt+√(2)dW_t+dZ_t,  
    t>0,

where the initial value X_0=x∈


ℝ, (W_t)_t≥0 is a standard Brownian motion, and (Z_t)_t≥0 is
an independent compound Poisson process with a unit jump intensity and a unit
mean exponential jump distribution. The two-sided exit quantities of this
generalized PEMP can also be solved using the approach described in Sections 6
and 7 of Jacobsen and Jensen <cit.>.

We define an integral kernel

    ψ_1(z)=e^z^2/2/z(z+1),   z∈
    
    
    ℂ
    .

Let Γ_i (  i=1,2) be small counterclockwise circles
around the simple poles μ_1=0 and μ_2=-1, respectively, and define
the linearly independent functions

    g_1(x)       :=1/2π√(-1)∫_Γ_1ψ_1(z)e^-xz
    dz=1,
    
    g_2(x)       :=1/2π√(-1)∫_Γ_2ψ_1(z)e^-xz
    dz=-e^x+1/2,

for x∈


ℝ. To find another linearly independent partial eigenfunction, we consider the
vertical line Γ_3={1+t√(-1),t∈


ℝ
} and define

    g_3(x):=1/2π√(-1)∫_Γ_3ψ_1(z)e^-xz
    dz.

Next we derive an explicit expression for g_3(x). We know from
(<ref>) that lim_x→∞g_3(x)=0 and g_3 is
continuously differentiable with

    g_3^'(x)=-1/2π√(-1)∫_Γ_3e^z^2/2/z+1e^-xzdz.

Notice that the bilateral Laplace transform functions (e.g., Chapter VI of
<cit.>) of a standard normal random variable U_1 and an independent
unit mean exponential random variables U_2 are given respectively by

    ∫_-∞^∞e^-zy·1/√(2π)e^-y^2/2
    dy=e^z^2/2,  ∫_0^∞e^-zy·
    e^-ydy=1/z+1,

for all complex z such that (z)≥0. Hence, the bilateral Laplace
transform of the density function of U_1+U_2, i.e.,

    ∫_0^∞1/√(2π)e^-(x-y)^2/2e^-y
    dy

is given by e^z^2/2/(z+1) for all complex z such that
(z)≥0. Since the right hand side of (<ref>) is just the
Bromwich integral for the inversion of the bilateral Laplace transform
-e^z^2/2/(z+1), evaluated at -x, we deduce that

    g_3^'(x)=-∫_0^∞1/√(2π)e^-(x+y)^2
    /2e^-ydy.

It follows that

    g_3(x)=-∫_x^∞g_3^'(y)dy=1-∫_0^∞N(x+y)e^-ydy.

where N(·) is the cumulative distribution function of standard normal distribution.

For any fixed -∞<u<v<∞, we define a matrix-valued function

    (M_i,k(u,v))_1≤ i,k≤3:=
    [                    1               g_1(v)               g_1(u);             ve^v+1/2               g_2(v)               g_2(u); 1-∫_0^∞N(v+y)ye^-ydy               g_3(v)               g_3(u) ]
    ,

where the first row is computed according to

    M_i,1(u,v)=1/2π√(-1)∫_Γ_iψ_0(z)/z+1e^-vzdz.

Notice that M_3,1(u,v) can be calculated in the same way as g_3(x). We
also denote by (N_k,j(u,v))_1≤ k,j≤3 the inverse of (M_i,k
(u,v))_1≤ i,k≤3.

By Eq. (46) and a generalized Eq. (48) of Jacobsen and Jensen <cit.>
(with ζ=s=0 and ρ≥0), we obtain the linear system of equations

    (c_1,c_2,c_3)(M_i,k)=(  C/ρ+1
    ,f(v),f(u))  ,

where C is a constant specified later, and f(x) could stand for
any of B_1^(0)(x;u,v), B_2^(0,ρ)(x;u,v), or C^(0,0)(x;u,v)
and has the representation

    f(x)=∑_i=1^3c_ig_i(x),   x∈ u,v].

By letting (1) C=f(u)=0 and f(v)=1, (2) C=1 and
f(v)=f(u)=0, (3) C=f(v)=0 and f(u)=1, for any ρ≥0 and
z>0, and solving the linear system (<ref>), we respectively obtain

    B_1^(0)(x;u,v)       =∑_i=1^3N_2,i(u,v)g_i(x),
    
    B_2^(0,ρ)(x;u,v)       =1/1+ρ∑_i=1^3N_1,i
    (u,v)g_i(x),   B_2^(0)(x,dz;u,v)=e^-z∑_i=1^3
    N_1,i(u,v)g_i(x)dz,
    
    C^(0,0)(x;u,v)       =∑_i=1^3N_3,i(u,v)g_i(x).

Furthermore, this implies

    b_a,1^(0)(x)       =-∑_i=1^3D_2,1(x-a,x)g_i(x),
    
    b_a,2^(0)(x,dz)       =e^-z(  ∑_i=1^3D_1,i
    (x-a,x)g_i(x))  ,
    
    c_a^(0,0)(x)       =∑_i=1^3D_3,i(x-a,x)g_i(x),

where we denote D_k,j(u,v)=∂/∂ vN_k,j(u,v).

In Figure 2 below, we plot h(x)=ℙ_x{M_τ_a≤ K} by
numerically solving the integral equation (<ref>) using
.





§ ACKNOWLEDGMENTS


The authors would like to thank two anonymous referees for their helpful
comments and suggestions. Support from grants from the Natural Sciences and
Engineering Research Council of Canada is gratefully acknowledged by David
Landriault and Bin Li (grant numbers 341316 and 05828, respectively). Support
from a start-up grant from the University of Waterloo is gratefully
acknowledged by Bin Li, as is support from the Canada Research Chair Program
by David Landriault.






§ APPENDIX




 §.§ Proof of Lemma <ref>


We define ψ _n(z)=inf_m≥ nϕ _m(z) for z∈ S. Further,
we define ψ_n(z)=lim inf_w→ zψ _n(w)
which is lower semi-continuous (see, e.g., Lemma 5.13.4 of Berberian <cit.>). Note that ψ_n is increasing in n, and by the definition of ψ_n, we have 

    lim_n→∞ψ_n(z)=   lim_n→∞lim_r↓ 0inf_w∈ (z-r,z+r)inf_m≥ nϕ
    _m(w)
    
    =   lim_n→∞lim_r↓ 0inf_m≥ n,w∈ (z-r,z+r)ϕ
    _m(w)≡lim inf_n→∞ ,w→ zϕ _n(w),

where the second equality is because there is no ambiguity in switching the order of two infimums. 
By the monotone convergence theorem, we have 

    ∫_Slim inf_n→∞ ,w→ zϕ _n(w)d
    μ (z)=lim_n→∞∫_Sψ_n(z)d
    μ (z).

By Portmanteau theorem of weak convergence and the fact that ψ_n(z) is nonnegative and lower semi-continuous, it follows that 

    ∫_Sψ_n(z)dμ (z)≤lim inf_m→∞∫_Sψ_n(z)dμ _m(z)

for any n∈

ℕ. Moreover, since ψ _n(z) is monotone increasing in n, we have

    lim inf_m→∞∫_Sψ_n(z)dμ
    _m(z)≤lim inf_m→∞∫_Sψ_m(z)
    dμ _m(z).

By (<ref>)-(<ref>),

    ∫_Slim inf_n→∞ ,w→ zϕ _n(w)d
    μ (z)≤lim inf_m→∞∫_Sψ_m(z)
    dμ _m(z)≤lim inf_m→∞∫_Sϕ _m(z)
    dμ _m(z),

where the last inequality is due to ψ_m(z)≤ψ
_m(z)≤ϕ _m(z). 






Suppose that {ϕ _n}_n∈
	
	ℕ is uniformly bounded by K>0, by applying (<ref>) to {K-ϕ
_n}_n∈
	
	ℕ, we obtain 

    Kμ (S)-∫_Slim sup_n→∞ , w→ zϕ _n(w)dμ
    (z)    =∫_Slim inf_n→∞, w→ z(K-ϕ _n(w))dμ (z)
    
       ≤lim inf_n→∞∫_S(K-ϕ _n(z))dμ
    _n(z) 
        =Klim inf_n→∞μ _n(S)-lim sup_n→∞∫_Sϕ _n(z)dμ _n(z).

Therefore, inequality (<ref>) follows immediately by the weak
convergence of μ _n and μ (S)<∞.



 §.§ Assumption verification for Example <ref>



Consider the PMEP (<ref>) with a continuous jump size
distribution F(·). For q,s≥0 and 0<u_0<x_0<v_0, we have

    lim_(u,v)↓(u_0,v_0)g(x_0;u,v)=lim_(x,u)↑
    (x_0,u_0)g(x;u,v_0)=g(x_0,u_0,v_0),

where the function g(x;u,v) is any of the following three functions:
B_1^(q)(x;u,v), B_2^(q,s)(x;u,v) and C^(q,s)(x;u,v).



Note that the condition 0<u_0<x_0<v_0 is to ensure the process X remains positive before exiting these finite intervals, which further implies X is upward regular and creeps upward.
We limit our proof to

    lim_(u,v)↓(u_0,v_0)B_1^(q)(x_0;u,v)=B_1^(q)
    (x_0;u_0,v_0).

The other results can be proved in a similar manner.
By the relationship v>v_0>u>u_0, we have

    | B_1^(q)(x_0;u_0,v_0)-B_1^(q)(x_0;u,v)|
       ≤|𝔼_x_0[  e^-qT_v_0^+1_{T_v_0
    ^+<T_u_0^-,X_T_v_0^+=v_0}]  -𝔼_x_0
    [  e^-qT_v^+1_{T_v^+<T_u^-,X_T_v^+=v,X_T_v_0
    ^+=v_0}]  |
         +ℙ_x_0{  v_0<X_T_v_0^+≤ v}  .

It is clear that the last term of (<ref>) vanishes as v↓
v_0 by the right-continuity of the distribution function of X_T_v_0
^+. Also,

    |𝔼_x_0[  e^-qT_v_0^+1_{T_v_0
    ^+<T_u_0^-,X_T_v_0^+=v_0}]  -𝔼_x_0
    [  e^-qT_v^+1_{T_v^+<T_u^-,X_T_v^+=v,X_T_v_0
    ^+=v_0}]  |
         =𝔼_x_0[  e^-qT_v_0^+1_{T_v_0^+<T_u
    ^-,X_T_v_0^+=v_0}]  -𝔼_x_0[
    e^-qT_v^+1_{T_v^+<T_u^-,X_T_v^+=v,X_T_v_0^+
    =v_0}] 
         +𝔼_x_0[  e^-qT_v_0^+1_{T_u^-<T_v_0
    ^+<T_u_0^-,X_T_v_0^+=v_0}] 
       ≤1-𝔼_v_0[  e^-qT_v^+1_{T_v^+<T_u
    ^-,X_T_v^+=v}]  +ℙ_x_0{  T_u^-
    <T_v_0^+<T_u_0^-}  .

Let ζ be the time of the first jump of the compound Poisson process Z
with jump rate λ>0.
Note that X will increase continuously up to time ζ as long as the initial value is positive. Since v>v_0>0, we have

    1-𝔼_v_0[  e^-qT_v^+1_{T_v^+<T_u^-
    ,X_T_v^+=v}]  ≤1-𝔼_v_0[  e^-qT_v^+
    1_{ζ>T_v^+}]  =1-(
    v/v_0)  ^-(q+λ)/μ.

By conditioning on X_T_u^--, one obtains

    ℙ_x_0{  T_u^-<T_v_0^+<T_u_0^-}   ≤∫_u^v_0ℙ_x_0{  X_T_u^--∈dy}ℙ{  y-u<J≤ y-u_0}
       ≤max_u_0≤ y≤ v_0(  F(y-u_0)-F(y-u))  .

Since F(·) is continuous, and hence uniformly continuous for
y∈0,v_0-u_0], it follows that the right-hand side of (<ref>)
vanishes as u↓ u_0. From (<ref>)–(<ref>), we conclude
that (<ref>) holds.

Note that although (<ref>) only uses the continuity of F on [0,∞),
the proof for C^(q,s)(x;u,v) will use the continuity of F on
(-∞,0].


 Assumptions (A1)-(A3) hold for the piecewise
exponential Markov process (<ref>) with a continuous jump size
distribution F(·) and initial value X_0≥ a.



For 0<u<x<v, by the strong Markov property, we have

    B_1^(q)(x;u,v)       =𝔼_x[  e^-qT_v^+1_{T_v
    ^+<T_u^-,X_T_v^+=v,ζ>T_v^+}]  +𝔼
    _x[  e^-qT_v^+1_{T_v^+<T_u^-,X_T_v^+
    =v,ζ<T_v^+}] 
         =(  v/x)  ^-(q+λ)/μ+λ∫_0^1/μlnv/xe^-(q+λ)tdt∫_u-xe^μ t
    ^v-xe^μ tB_1^(q)(xe^μ t+w;u,v)F(dw).

By Lemma <ref>, Eq. (<ref>), and the
dominated convergence theorem, it is straightforward to verify that Assumption
(A1) holds and for x>a,

    b_a,1^(q)(x)=q+λ/μ x-λ/μ x∫_-a
    ^0B_1^(q)(x+w;x-a,x)F(dw).

Note that we require x>a as otherwise x+w in the above equation could be negative for w∈(-a,0), and then Lemma <ref> does not apply.
Obviously, ∫_x^yb_a,1^(q)(w)dw<∞ for all
0<x<y<∞. Similarly, by conditioning on the first jump of Z,
for 0<u<x<v,

    B_2^(q)(x,dz;u,v)       =λ∫_0^1/μlnv/xe^-(q+λ)tF(v-xe^μ t+dz)dt
         +λ∫_0^1/μlnv/xe^-(q+λ)t
    dt∫_u-xe^μ t^v-xe^μ tB_2^(q)(xe^μ
    t+w,dz;u,v)F(dw),

and

    C^(q,s)(x;u,v)=λ∫_0^1/μlnv/xe^-(q+λ
    )tdt∫_-∞^v-xe^μ tC^(q,s)(xe^μ t
    +w;u,v)F(dw),

where it is understood that C^(q,s)(xe^μ t+w;u,v)=e^s(xe^μ t
+w-u) for w<u-xe^μ t. One can verify from Lemma <ref> and the
dominated convergence theorem that Assumptions (A2) and (A3)
hold, and for x>a,

    b_a,2^(q)(x,dz)=λ/μ xF(dz)+λ/μ x∫_-a^0B_2^(q)
    (x+w,dz;x-a,x)F(dw),

and

    c_a^(q,s)(x)=λ/μ x∫_-∞^0C^(q,s)
    (x+w;x-a,x)F(dw).

This ends the proof.




 §.§ Assumption verification for Example <ref>


Let U be the continuous component of X, which is a linear diffusion
process with the infinitesimal generator ℒ_U=1/2
σ^2(y)d^2/dy^2+μ(y)d
/dy. It is well-known that, for any q>0, there exist two
independent and positive solutions, denoted as ϕ_q^±(y), to the
Sturm-Liouville equation

    ℒ_Uϕ_q^±(y)=qϕ_q^±(y),

where ϕ_q^+(·) is strictly increasing and ϕ_q^-(·)
is strictly decreasing. By the Lipschitz assumption on μ(·) and
σ(·), it follows from the Schauder estimates (e.g., Theorem 6.14 of
Gilbarg and Trudinger <cit.>) of Eq. (<ref>) that ϕ_q^±(·)∈ C^2,α(Ω̅) for any α∈(0,1] and any
compact set Ω̅⊂


ℝ. Interested readers can refer to Section 4.1 of Gilbarg and Trudinger
<cit.> for more detail on the Hölder space C^2,α(Ω̅).

We denote the first hitting time of U to level z∈


ℝ by H_z=inf{t>0:U_t=z}. It is well-known that, for u≤ x≤ v,

    𝔼_x[  e^-qH_u1_{  H_u<H_v}]
    =f_q(x,v)/f_q(u,v)  and  𝔼_x[
    e^-qH_v1_{  H_v<H_u}]  =f_q(u,x)/f_q(u,v),

where f_q(x,y):=ϕ_q^+(x)ϕ_q^-(y)-ϕ_q^+(y)ϕ_q
^-(x). Note that f_q(x,y) is strictly decreasing in x and strictly
increasing in y with f_q(x,x)=0. In particular, for u≤ x≤ v, we
have

    𝔼_x[  e^-qH_u]  =ϕ_q^-(x)/ϕ
    _q^-(u)  and  𝔼_x[  e^-qH_v]
    =ϕ_q^+(x)/ϕ_q^+(v).

For 𝐞_q an independent exponential random variable with mean
1/q<∞, the q-potential measure of U is given by

    r_q(x,y):=1/qℙ_x{  U_𝐞_q∈dy}  /dy={[ 2/qσ^2(y)ϕ_q^+(x)ϕ_q^-(y)/f_q,1
    (y,y),   x≤ y,;  2/qσ^2(y)ϕ_q^+(y)ϕ_q^-(x)/f_q,1
    (y,y),   x>y, ].

where f_q,1(x,y):=∂/∂ xf_q(x,y). Furthermore, the
q-potential measure of U killed on exiting the interval [u,v], for
u≤ x,y≤ v, is given by

    θ^(q)(x,y;u,v)       :=1/qℙ_x(  U_𝐞
    _q∈dy,𝐞_q<H_u∧ H_v)  /d
    y
         =r_q(x,y)-f_q(x,v)/f_q(u,v)r_q(u,y)-f_q(u,x)/f_q(u,v)r_q(v,y).



The next lemma is an analogy of Lemma <ref>. Thanks to the diffusion
term in the jump diffusion model (<ref>), we now allow for the presence of
atoms in the jump intensity measure ν(·).


Consider the jump diffusion model (<ref>). For q,s≥0 and
u_0<x_0<v_0, we have

    lim_(u,v)↓(u_0,v_0)g(x_0;u,v)=lim_(x,u)↑
    (x_0,u_0)g(x;u,v_0)=g(x_0,u_0,v_0),

where g(x;u,v) is any of the following functions: B_1^(q)(x;u,v),
B_2^(q,s)(x;u,v) and C^(q,s)(x;u,v).



We can follow the same proof as Lemma <ref> except for the term
ℙ_x_0{  T_u^-<T_v_0^+<T_u_0^-}
in (<ref>), which will be handled distinctly here. We have X_t=U_t
a.s. for t<ζ, where ζ is the first time a jump occurs which
follows an exponential distribution with mean 1/λ=1/ν(ℝ)>0.
For any u_0<u<x_0<v_0, by (<ref>) and (<ref>), we have

    ℙ_x_0{  T_u^-<T_v_0^+<T_u_0^-}   ≤ℙ_u{  T_v_0^+<T_u_0^-}
         =ℙ_u{  T_v_0^+<T_u_0^-,ξ>T_v_0
    ^+}  +ℙ_u{ξ≤ T_v_0^+<T_u_0
    ^-}
       ≤𝔼_u[  e^-λ H_v_01_{  H_v_0
    <H_u_0}]  +1-𝔼_u[  e^-λ H_u_0
    ] 
         =f_q(u_0,u)/f_q(u_0,v_0)+1-ϕ_q^-(u)/ϕ_q^-(u_0).

Therefore, it follows that lim_u↓ u_0ℙ_x_0{
T_u^-<T_v_0^+<T_u_0^-}  =0 by f_q(u_0,u_0
)=0.


Assumptions (A1)-(A3) hold for the jump diffusion model
(<ref>).



By the strong Markov property, (<ref>) and (<ref>), for u<x<v, it
follows that

    B_1^(q)(x;u,v)
         =𝔼_x[  e^-qT_v^+1_{T_v^+<T_u^-,T_v
    ^+=v,ζ>T_v^+}]  +𝔼_x[  e^-qT_v^+
    1_{T_v^+<T_u^-,T_v^+=v,ζ<T_v^+}] 
         =𝔼_x[  e^-(q+λ)H_v1_{H_v<H_u}]
    +∫_u^v𝔼_x[  e^-qζ1_{ζ<H_u∧
    H_v,U_ζ∈dy}]  ∫_ℝB_1^(q)
    (y+γ(y,w);u,v)ν(dw)/λ
         =f_q+λ(u,x)/f_q+λ(u,v)+∫_u^vθ
    ^(q+λ)(x,y;u,v)dy∫_ℝB_1^(q)(y+γ
    (y,w);u,v)ν(dw),

where it is understood that B_1^(q)(y+γ(y,w);u,v)=0 if
γ(y,w)>v-y or γ(y,w)<u-y. By Lemma <ref>, the dominated
convergence theorem, and the identity f_q+λ(u,v)=-f_q+λ
(v,u), we can verify that Assumption (A1) holds with

    b_a,1^(q)(x)=-f_q+λ,1(x-a,x)/f_q+λ(x-a,x)
    -∫_x-a^xθ̃_a^(q+λ)(x,y)dy∫
    _ℝB_1^(q)(y+γ(y,w);x-a,x)ν(dw),

where we write θ̃_a^(q+λ)(x,y):=-f_q+λ
,1(x-a,x)/f_q+λ(x-a,x)r_q+λ(x,y)-r_q+λ,1
(x,y)+f_q+λ,1(x,x)/f_q+λ(x-a,x)r_q+λ(x-a,y)
and r_q+λ,1(x,y):=∂/∂ xr_q+λ(x,y). The
integrability of b_a,1^(q)(·) follows from the continuity of the
ϕ_q^+(·) and ϕ_q^-(·).

Similarly, by the strong Markov property of X, (<ref>) and
(<ref>), we have

    B_2^(q)(x,dz;u,v)=∫_u^vθ^(q+λ)
    (x,y;u,v)dy∫_ℝB_2^(q)(y+γ(y,w),d
    z;u,v)ν(dw),

and

    C^(q,s)(x;u,v)=f_q+λ(x,v)/f_q+λ(u,v)+∫_u
    ^vθ^(q+λ)(x,y;u,v)dy∫_ℝC^(q,s)
    (y+γ(y,w);u,v)ν(dw).

One can verify from Lemma <ref> that Assumptions (A2) and
(A3) hold with

    b_2,a^(q)(x,dz)=∫_x-a^xθ̃_a^(q+λ
    )(x,y)dy∫_ℝB_2^(q)(y+γ(y,w),d
    z;x-a,x)ν(dw),

and

    c_a^(q,s)(x)=-f_q+λ,1(x,x)/f_q+λ(x-a,x)+∫
    _x-a^xθ̃^(q+λ)(x,y)dy∫_ℝ
    C^(q,s)(y+γ(y,z);x-a,x)ν(dw).

This completes the proof.


13pt

99                                                                                               


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