On the the successive passage times of certain one-dimensional diffusions
Mario Abundo Maria Beatrice Scioscia Santoro




On the successive passage times of certain one-dimensional diffusions.
    Mario Abundo^*, Maria Beatrice Scioscia SantoroDipartimento di Matematica, Università  “Tor Vergata”, via della Ricerca Scientifica, I-00133 Rome, Italy.
Corresponding author E-mail: abundo@mat.uniroma2.it

    
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We show in detail some results, outlined in a  previous paper regarding the case of Brownian motion (BM),
about the distribution of the nth-passage time of a one-dimensional diffusion obtained by a space or time transformation of
BM, through a constant barrier a. Some explicit examples are reported.




Keywords: First-passage time, Second-passage time, one-dimensional diffusion

Mathematics Subject Classification: 60J60, 60H05, 60H10.



§ INTRODUCTION

This is a continuation of <cit.>, where we studied excursions of
Brownian motion with drift, and we found
explicitly the distribution of  the nth-passage time of Brownian motion (BM)
through a linear boundary.
In the present paper, by using the results of <cit.>, we study in depth
the nth-passage time of a one-dimensional diffusion process X(t), which is obtained
from BM by a space transformation or a time change. Particularly, focusing on
the second passage-time, we
find explicit formulae for the density of the second-passage time, and we calculate
its Laplace transform. We use
numerical computation and/or simulation, whenever analytical expressions cannot be achieved,
and we show graphically the results in a number of figures.
 The successive-passage times of a diffusion X(t) through a
boundary S(t) are related to the excursions of Y(t):=X(t) -
S(t); indeed, when Y(t)  is entirely positive or entirely
negative on the time interval (s,u), it is said that it is an
excursion of Y(t). Excursions have interesting applications in
Biology, Economics, and other applied sciences (see e.g.
<cit.>, <cit.>). They are also related to the
last passage time of a diffusion through a boundary; actually,
last passage times of continuous martingales play an important
role in Finance, for instance, in models of  default risk (see
e.g. <cit.>, <cit.>). In the special case when
X(t) is BM, its  excursions  follow the arcsine law, namely the
probability that BM has no zeros in the time interval (s,u) is
2/πarcsin√(s/u) (see e.g. <cit.>).
Really, by using Salminen's formula for the last passage time of
BM through a linear boundary (see <cit.>), in
<cit.> we found the law of the excursions of drifted
BM, and  we derived the distribution of the nth passage time of
BM through a linear boundary; in this article, as we said, our aim is
to obtain analogous results for transformed BM through a constant boundary. The paper is organized as
follows: in Section 2, we recall the results of
<cit.>, concerning the nth-passage time of BM
through a linear boundary, in Section 3 we extend these results to
certain diffusions X(t), studying also the Laplace transform of
the second-passage time of X(t). Section 4 reports some explicit examples,
while Section 5 is devote to conclusions and final remarks.



§ PRELIMINARY RESULTS ON THE NTH-PASSAGE TIME OF BROWNIAN MOTION 

The first-passage time of BM  through the linear boundary S(t)= a + bt, when starting
from x ,  is defined by τ ^B_1 (S|x) = inf{ t >0: x +B_t = a + bt } , that will be denoted in this section  by
τ _1 (x), dropping, for simplicity,  the superscript B which refers to BM and the dependence on S;  we recall
the Bachelier-Levy formula for the distribution of τ_1(x):

    P(τ_1 (x) ≤ t ) = 1 - Φ((a -x) / √(t) + b √(t) )
     + exp(-2b (a-x)) Φ(b √(t) -(a-x)/ √(t) ),  t >0

where Φ (y)= ∫ _ - ∞ ^y ϕ(z) dz , with
ϕ (z) = e^ - z^2/2/ √(2 π),
is the cumulative distribution function of the standard Gaussian variable.
If (a -x)b > 0 , then P(τ _1 (x) < ∞ )= e^-2b(a-x), whereas, if
(a -x)b ≤ 0 ,  τ _1 (x)  is finite with probability one and it has the Inverse Gaussian density,
which is non-defective
(see e.g. <cit.>, <cit.>):

    f_ τ _1 (t) = f_ τ _1 (t |x) =  d/dt  P(τ _1 (x) ≤ t) =  |a-x| /t^ 3/2 ϕ (  a +bt -x /√(t) ) ,
     t >0;

moreover, if b ≠ 0, the expectation of τ _1 (x) is finite, being
E(τ _1(x)) = |a-x| /|b| . The second-passage time of BM  through
S(t), when starting from x , is defined by
τ _2 (x) :=  τ ^B_2 (S|x) = inf{ t > τ _1(x): x +B_t = a + bt } , and generally, for n ≥ 2,  τ _n (x) = inf{ t > τ _n-1(x): x +B_t = a + bt } denotes the nth-passage
time of BM through S(t). Now, we assume  that b ≤ 0 and x < a , or b ≥ 0 and x > a, so that  P( τ _1(x) < ∞ )=1.
For fixed t>0, we consider the last-passage time prior to t of BM, starting from x , through the boundary
S(t)=a + bt,
that is:

    λ _S ^t =
    sup{ 0 ≤ u ≤ t : x + B_u = S(u) }    if  τ_1(x) ≤ t 
    
    0     if  τ_1(x) > t .

Notice that  <cit.> contains a slight imprecision in the definition of λ _S ^t , and the function there denoted by ψ _t (u), has
to be thought indeed as
the density  of λ _S ^t , conditional to the event {τ _1(x) ≤ t }; in other words, λ _S ^t is zero with
probability P( τ _1 (x) >t), and it takes values spread on the interval (0,t) with conditional density
ψ _t (u)= P(λ _S ^t ∈ du | τ _1(x) ≤ t ). However, this not affects the correctness of the results of <cit.>.
In fact, it holds (see <cit.>, <cit.>):

    ψ _t (u)= d/du P (λ _S ^t ≤ u | τ _1 (x) ≤ t ) =
    1/√(2 π u )exp (-b^2 u/2) ∫ _ - ∞ ^+ ∞ν _ x-a (t-u, S )  dx,  u ≤ t

where S (t) = S(t-u),  u ≤ t , and

    ν _x (v, S ) = exp{ - b(x-bt) - b^2 v /2 ) }|x-bt|/√( 2 π v^3)exp{ - (x-bt)^2 /2v }

Then, the following explicit formula is obtained (see <cit.>):

    ψ _t (u)=   e^ - b^2/2 u/π√(u(t-u)) [ e^ - b^2/2 (t-u) + b/2√( 2 π  (t-u))  ( 2 Φ ( b √(t-u) ) -1  )  ]  ,  0 <u <t

Notice that
ψ _t is independent of a;
if b=0, one gets
ψ _t (u) = 1/π√(u(t-u)) ,   0 <u <t ,
that is,  the arc-sine law with support in (0,t).

For z <t, the event {λ _S ^t ≤ z | τ_1(x) ≤ t } is nothing but the event { x+ B_u - S(u)
 has no zeros in the  interval (z,t) }.

In order to study the second-passage time, τ _2 (x), of x+ B_t through the linear boundary S(t)=a + bt ,
when x <a and b ≤ 0 (the case when b ≥ 0 and x > a can be studied in a similar way),
we set T_1(x)= τ _1(x) and T_2 (x) = τ_2(x) - τ _1 (x). It can be proved (see <cit.>) that
T_2 (x) is finite with probability one, only if b =0 .
Conditionally to τ _1 (x)=s, the event {τ _2 (x) > s+t } (t >0) , is nothing but the event { x + B_u - S(u)
 has no zeros in the interval  (s, s+t) } . Therefore, by using the above expression of ψ _t (u) one obtains

    P( τ _2 (x) > τ _1 (x) +t | τ _1 (x) =s ) =  P( λ _S ^s+t≤ s | τ _1 (x) ≤ s+t) = ∫ _0 ^s ψ _ s+t (y) dy

and so

    P( T_2(x) ≤ t | τ _1 (x) =s ) = 1 - ∫ _ 0 ^s ψ _s+t (y) dy .

Then:

    P( T_2 (x) ≤ t ) = 1- ∫ _ 0 ^+ ∞ f_τ _1 (s) ds ∫ _ 0 ^s ψ _s+t (y) dy .

By taking the derivative with respect to t, one gets the density of T_2(x):

    f_T_2 (t) = ∫ _ 0 ^+ ∞ f_τ _1 (s)   [ e^ - b^2 (s+t)/2 √(s)/π√(t) (s+t)  ]  ds .

Notice that ∫ _ 0 ^s ψ _s+t (y) dy is decreasing in t


and
f_T_2 (t) ∼  const / √(t), as t → 0^+.
Moreover, from (<ref>) it follows that, if b ≠  0, the distribution of T_2(x) is  defective

In fact (see <cit.>):

    P( T_2(x) = + ∞ ) = 2  sgn (b) E  ( Φ ( b √(τ_1(x)) ) - 1/2 ) >0,

where sgn (b) =

0     if  b=0 
  |b| / b     if  b ≠ 0
. Since the function g(s)= 2  sgn (b)  ( Φ ( b √( s) ) - 1/2 ) is concave, by  Jensen's inequality
one gets
P( T_2(x) = + ∞ ) ≤γ (b), where γ (b) = 2  sgn (b)  ( Φ ( b √(  |(a-x)/b|  ) ) - 1/2 ) .
Notice that  γ is an even function of b. On the contrary, if b=0, from (<ref>) we get that P(T_2(x) = + ∞ ) =0, that is, T_2(x) is a proper random variable,
and
also τ _2 (x) = T_2(x) + τ _1 (x) is finite with probability one;
precisely, by calculating the integral in (<ref>)
we have:

    P( T_2(x) ≤ t ) = ∫ _0 ^+ ∞2/πarccos√(s/s+t)|a-x| /√( 2 π) s^3/2 e^ - (a -x)^2 /2s  ds

then, taking the derivative with respect to t, the density of T_2(x) turns out to be:

    f_T_2 (t) = ∫ _ 0 ^+ ∞1/π (s+t) √(t)|a-x| /√( 2 π) s  e^ - (a -x)^2 /2s  ds .

In the Figure 1, taken from  <cit.>,
the plot of P(τ _2(x) = + ∞) = P(T _2(x) = + ∞),
as a function of b≤ 0,  is compared with the plot of its upper bound γ (b),  for a=1 and x=0.



In the Figure 2, taken from <cit.>, it is reported the probability density of T_2(x), obtained from (<ref>),
by calculating numerically the integral, for
x=0,  a=1, and various values of the parameter b ≤ 0.




As far as the expectation of  τ _2 (x) is concerned, it is obviously infinite for b ≠ 0 , while E(τ _1 (x))
is finite.
If b=0,  E( τ _1(x)) and E( τ _2(x))
are both infinite.
As for τ _1 (x) this is well-known, as for τ _2 (x) it derives from the fact that (see <cit.>):

    E(T_2 (x)) = ∫ _ 0 ^+ ∞ P( T_2(x) >t ) dt =


    ∫ _ 0 ^ + ∞ ds  2/π |a-x| /√( 2 π) s^3/2 e^ - (a -x)^2 /2s ∫ _ 0 ^ + ∞arcsin√(s/s+t) dt  = + ∞  .

By taking the derivative with respect to t in (<ref>),
one obtains the  density of T_2(x) conditional to τ _1(x)=s, that is:

    f_T_2 | τ _1 (t | s) = - d/dt∫ _0 ^s  ψ _ s +t  (y) dy = e^ - b^2 (s+t)/2√(s)/π (s+t) √(t) ,

and, for b=0:

    f_T_2 | τ _1 (t | s) = √(s)/π (s+t) √(t) .

Since τ _2 (x) = τ _1 (x) + T_2 (x), by the convolution formula, the density of τ _2 (x) follows:

    f_ τ _2 (t) = ∫ _ 0 ^t f_T_2 | τ _1 (t-s | s) f _ τ _1  (s) ds  = e^  - b^2 t/2/π t∫ _0 ^t |a -x| /√( 2 π) s √( t-s ) e^ - (a +bs-x)^2 /2s  ds .

Of course, the distribution of τ _2 (x)  is defective for b ≠ 0, namely
∫ _ 0 ^ + ∞ f_ τ _2 (t) dt = 1 - P( τ _2 (x)= + ∞) < 1 ,
since P( τ _2 (x)= + ∞) = P( T _2 (x)= + ∞) >0. If b=0, we obtain:

    f_ τ _2 (t) = 1/π t ∫ _0 ^t |a -x| /√( 2 π) s √( t-s ) e^ - (a -x)^2 /2s  ds ,

which is non-defective. In the Figure 3, we report the probability density of τ _2 (x), obtained from (<ref>) by calculating numerically the integral, for
x=0,  a=1, and various values of the parameter b ≤ 0. Although the shapes appear to be similar to that of the inverse Gaussian density
(<ref>), the density
of τ _2 (x) is more concentrated around its maximum. In the Figure 4, we report from <cit.> the comparison between the probability density of τ _2 (x) and the inverse Gaussian density, for a=1, b=0 and x=0.






The following holds (see <cit.>):
 
Let be
T_1(x) = τ _1 (x),  T_n (x)= τ  _n (x) - τ _ n-1 (x),   n =2, …  Then:

    P(T_1 (x) ≤ t ) = 2(1 - Φ(a -x / √(t)  )),


    P( T_n (x) ≤ t ) = 1 - ∫ _ 0 ^+ ∞ f_τ _ n-1 (s) ds ∫ _ 0 ^s ψ _s+t (y) dy,  n= 2, …

Moreover, the density of τ _n (x) is:

    f_ τ _n (t) = ∫ _ 0 ^t f_T_n | τ_n-1 (t-s | s) f _ τ _n-1 (s) ds ,

where f_τ _ n-1 and f_ T_n | τ _ n-1 can be calculated inductively, in a similar way, as done for f_τ _ 2
and f_T_2 | τ _1  . If b=0,  T_1(x),  T_2 (x), .... are finite with probability one.





We conclude this section, by summarizing the Laplace transforms  of the first and second passage-time of B_t through the boundary S(t)= a+ bt: ∙ the Laplace transform (LT) of τ _1 (x) is the LT of the inverse Gaussian density:

    ψ(λ) = E  [ e^  - λτ_1(x)  ] =  e^-(a - x)(√(b^2 + 2λ) - b),  λ >0 ;

∙ the Laplace transform (LT) of τ _2 (x) is:

    ψ̃(λ) = E [e^- λτ_2(x)] = ∫_0^+∞f_τ_2(t)e^-λ t dt =
    ∫_0^+∞e^-λ t[e^-b^2/2t/π t∫_0^t |a - x|e^-(a + bs - x)^2/2s/√(2π)·
    s √(t - s) ds] dt,


In the Figure <ref>, we compare the shape of the LT of the inverse Gaussian density with the LT  of the second-passage time  τ_2(x)
of BM through the boundary S(t)=a + bt , for a=1, b=0 and x=0.






§ THE NTH-PASSAGE TIME OF A TRANSFORMED BROWNIAN MOTION

In this section we will extend the previous results  to a one-dimensional, time homogeneous diffusion X(t), obtained from BM by a space
transformation or
a time change. We start with diffusions conjugated to BM.



 §.§ The nth-passage time of a diffusion conjugated to BM

Let be X(t) a  one-dimensional, time-homogeneous diffusion process, which is driven by the SDE

    dX(t) = μ (X(t)) dt + σ (X(t)) d B_t,  X(0)= x ,

where the coefficients μ ( · ) and σ ( ·) satisfy the usual conditions for the existence and uniqueness of
the solution (see e.g. <cit.>).We recall that X(t) is said to be conjugated  to BM (see <cit.>), if there exists an increasing function
v with v(0) = 0, such that:

    X(t) = v^-1(B_t + v(x)),   t ≥ 0.

Examples of diffusions conjugated to BM are (see also <cit.>):
(i) the Feller process or Cox-Ingersoll-Ross (CIR) model, which is driven by the SDE

    dX(t) = 1/4 dt + √(X(t)  0) dB_t,      X(0) = x,
 and is
conjugated to BM via the function v(x) = 2√(x), i.e.
X(t) = 1/4(B_t + 2√(x))^2; (ii) the Wright and Fisher-like process, which is driven by the SDE

    dX(t) = (1/4 - 1/2 X(t)) dt + √(X(t)(1 - X(t))  0)  dB_t,      X(0) = x ∈ [0, 1],

and is conjugated to BM via the function v(x) = 2  arcsin√(x), i.e.

    X(t) = sin^2(B_t/2 + arcsin√(x) ).
 Suppose now that X(t) is conjugated to BM via the function v;
let τ _1 (a|x) = inf{ t>0: X(t) =a } be
the FPT of X(t) through
the level a >x ;
as easily seen, one has
τ _1 (a|x) =  τ _1^B(v(a) |v(x)),
where τ _1^B(α |y), denotes the FPT of BM, starting from y < α , through the level α.
Then, from (<ref>), one gets the density of τ _1 (a|x):

    f_τ_1(t) =  v(a) - v(x)/t^3/2ϕ(v(x) - v(a)/√(t)),

implying that
E [ τ _1 (a|x) ] = +∞.
For x < a, one has:

    τ _2 (a|x)  = inf{t > τ _1 (a|x): X(t) = a } =
    inf{t > τ _1 (a|x): B_t = v(a) - v(x) } =  τ _2^B (v(a)|v(x)) ,

and

    τ _n (a|x) = inf{t > τ_n - 1(a|x) : B_t = v(a) - v(x) } = τ _n^B (v(a)|v(x)) ,

where the superscript B refers to BM, namely τ_n^B(a|x),  n=2, … is the nth-passage time of BM through the level a,
when starting from x <a .
By  using the same notation of the previous section, setting T_1(x) = τ_1 (a |x) = τ _1 ^B (v(a)|v(x)),   T_2(x) = τ _2 (a|x) - τ _1 (a|x) =
τ _2 ^B (v(a)|v(x)) - τ _1 ^B (v(a)|v(x)),
and
using (<ref>) with b=0, we get

    P(T_2(x) ≤ t) = 2/π∫_0^+∞f_τ_1(s) arccos√(s/s + t) ds.

Thus, the density of T_2 is

    f_T_2(t)= ∫_0^+∞
    f_τ_1(s)  √(s)/π√(t) (s + t) ds  = ∫_0^+∞v(a) - v(x)/s^3/2e^-(v(x) -
    v(a))^2/2s/√(2π) √(s)/π√(t)(s + t) ds


    = v(a) - v(x)/π√(2π)∫_0^+∞1/(s +
    t)√(t) 1/s  e^-(v(x) - v(a))^2/2s ds.

The expectation of τ_2 (a|x) is infinite,
because E  [T_2(x) ] = +∞ .
From (<ref>) with b=0, we get

    f_T_2 τ_1(t s) = -d/dt∫_0^sψ_s + t(y) dy = √(s)/π(s + t)√(t).

Moreover, from (<ref>) (b=0):

    f_τ_2(t) = ∫_0^tv(a) - v(x)/π ts√(t -
    s)√(2π)· e^-(v(x) - v(a))^2/2s ds.

Setting T_n (x)= τ_n(a|x) - τ_n-1 (a|x), and
using Proposition <ref>, we get

    P (T_1(x) ≤ t) = 2(1 - Φ(v(a) - v(x)/√(t))),


    P (T_n(x) ≤ t ) = 1 -
    ∫_0^+∞f_τ_n - 1(s) ds∫_0^sψ_s + t(y) dy,    n = 2, ….

The density of τ_n (a|x) is:

    f_τ_n(t) = ∫_0^t f_T_n | τ_n - 1(t - s|  s)· f_τ_n - 1(s) ds,

where f_τ_n - 1 and f_T_n |  τ _n - 1 can be calculated inductively, in a similar way, as
done for f_τ _2 and f_T_2 | τ_1. In conclusion, T_1(x), T_2(x), … are finite with probability one.



 §.§ The nth-passage time of time-changed Brownian motion

The previous arguments can be applied  also to time-changed BM, namely

    X(t) = x + B(ρ(t)),

where ρ(t) ≥ 0 is an increasing, differentiable function of t > 0, with ρ(0) = 0.
Such kind of diffusion process X(t) is a special case of Gauss-Markov process; in particular the form (<ref>)
is taken by certain integrated
Gauss-Markov processes (see <cit.>, <cit.>).
Let us consider the constant barrier S = a, and x <a; then, the FPT of X(t) through a is
τ _ 1 (a|x) = ρ^-1( τ_1 ^B (a|x)),
where τ _1 ^B (a|x) denotes the first-passage time of BM, starting from x, through the barrier a.
Thus, the density of τ _ 1 (a|x) is :

    f_τ_1(t) = f_τ _1^B (ρ(t))  ρ'(t) = |a - x|/ρ(t)^3/2ϕ(a - x/√(ρ(t))) ρ'(t).

The expectation is:

    E [τ _ 1 (a|x)] = ∫_0^+∞t   f_τ_1(t) dt = ∫_0^+∞ t |a - x|·ρ'(t)/ρ(t)^3/2e^-(x - a)^2/2ρ(t)/√(2π) dt

Suppose that
ρ(t) ∼ c· t^α , as t → + ∞ ; then, as easily seen, for α >2 the integral converges,
namely E [τ _ 1 (a|x)] < +∞,
unlike the case of BM, for which the expectation of FPT is infinite.
One has:

    τ_2(a|x) = ρ^-1(τ _2 ^B (a|x))   with τ _2 ^B (a|x) = inf{s > τ _1 ^B (a|x) : x+B_s = a },

…………  ,

    τ_n(a|x) =
    ρ^-1(τ _n ^B (a|x))   with τ _n ^B (a|x) = inf{s > τ _n-1 ^B (a|x) : X+ B_s = a };


    T_1(x) = τ_1(a|x) = ρ^-1(τ _1 ^B (a|x)),   T_2(x) = τ_2(a|x) - τ_1(a|x) =
    ρ^-1(τ _2 ^B (a|x)) - ρ^-1(τ _1 ^B (a|x)).

It results

    P(T_2(x) ≤ t) = ∫_0^+∞ P(ρ^-1(τ _2 ^B (a|x)) ≤ t + s)
    f_ρ^-1(τ _ 1^B (a|x))(s) ds ,

and so the density of f_T_2 is

    f_T_2(t) = ∫_0^+∞f_τ _2 ^B (ρ(t + s)) ρ'(t + s) f_τ_1(s) ds
    = ∫_0^+∞f_τ _2 ^B(ρ(t + s)) ρ'(t + s) f_τ _1 ^B(ρ(s)) ρ'(s) ds.


Moreover,

    P(T_2(x) = +∞) = lim_t → +∞ P (T_2(x) > t)


    = 1 - ∫_0^+∞f_τ_1(s) [lim_t → +∞P (τ _2 ^B (a|x) ≤ρ(t + s))]  ds .

Therefore, by using (<ref>):

    P(T_2(x) = +∞) = 1 - ∫_0^+∞f_τ_1(s)
    [lim_t → +∞∫_0^ρ (t+s)f_τ_2^B(u) du ] ds


    =  1 - ∫_0^+∞f_τ_1(s)[lim_t → +∞∫_0^+∞[1/π u∫_0^ρ(t + s)|a - x| e^-(a - x)^2/2y/√(2π) y √(u - y) dy
    ] du] ds .

Moreover

    E[T_2(x)] = ∫_0^+∞ P (T_2(x) > t) dt


    = ∫_0^+∞[1 -
    ∫_0^+∞ P (τ_2 ^B(a|x) ≤ρ(t + s)) f_τ_1(s) ds],

which, unfortunately, it is impossible to calculate, due to the complexity of the integral.The density of τ _2 (a|x) is

    f_τ_2(t)  =
    d/dtP(τ_2 ^B (a|x) ≤ρ(t)) = f_τ_2^B(ρ(t))·ρ'(t)


    = ρ'(t)/πρ(t)∫_0^ρ(t)|a - x|/√(2π) s √(ρ(t) - s) e^-(a -
    x)^2/2s ds.

By using that
T_n(x) = τ_ n(a|x) - τ _n - 1(a|x) =ρ^-1(τ _n^B (a|x)) - ρ^-1(τ_n - 1^B (a|x)),
and conditioning with respect to τ_ n - 1(a|x) = s, one gets:

    P(T_n(x) ≤ t) = ∫_0^+∞ P(ρ^-1(τ _n ^B(a|x)) ≤ t + s) f_τ_n - 1(s) ds .

Thus, the density of T_n (x) is:

    f_T_n(t) = ∫_0^+∞d/dt P(τ _n ^B(a|x) ≤ρ(t + s)) f_τ_n - 1(s) ds =
    ∫_0^+∞f_τ_n ^B(ρ(t + s))ρ'(t + s) f_τ^n - 1(s) ds ,

while the density of τ_ n(a|x) is :

    f_τ_ n(t) = d/dt P (τ_n ^B (a|x) ≤ρ(t))


    =
    f_τ_n^B(ρ(t))ρ'(t) = ∫_0^ρ(t)f_T_n^B | τ_n - 1,^B(ρ(t) - s | s) f_τ_ n - 1^B(s) ds ·ρ'(t),

where f_T_n^B | τ_n - 1 ^B and f_τ_n - 1 ^B can be calculated inductively.



§ SOME EXAMPLES

Now, we will study  two interesting cases of time-changed BM, with regard to successive-passage times: B(t^3/3) and the Ornstein-Uhlenbeck (OU) process.


1)  B( t^3 /3) and integrated BM. Let be

    X(t) = ∫_0^t B_s ds

the integrated BM, which has many important application (see e.g. <cit.>, <cit.>). It is a Gaussian process with mean zero and covariance function Cov(X(t), X(s)) = s^2 (t/2 - s/6),  s ≤ t (see e.g. <cit.>).
Thus, its variance is Var (X(t)) = t^3/3,
and X(t) has the same distribution as B(t^3/3).
Motivated from this, we study the successive-passage time of B(t^3/3) through a constant barrier a.
Since the starting point is zero, we consider

    τ _1 (a|0)= inf{t > 0 : X(t) = a } = inf{t > 0 : B (t^3 / 3  ) = a},


    τ _2(a|0) = inf{t > τ_ 1(a|0) : B  (t^3/ 3  ) = a },

and, as before,

    T_1(x) = τ_1(a|0),  T_2(x) = τ_ 2(a|0) - τ_ 1(a|0).

Then, by using (<ref>), we get the density of τ_1 (a|0):

    f_τ_1(t) = |a - x|/(t^3/3)^3/2·ϕ(a - x/√(t^3/3))  t^2 =
    3√(3) |a - x|/t^5/2 e^-3(a - x)^2/2t^3/√(2π) .







In the Figure <ref> we report the FPT density of B(t^3/3) obtained from (<ref>),
for various values of the barrier: a = 1,  a = 2,   a = 3.
The expectation of τ_ 1(a|0) is obtained by (<ref>):

    E[τ_1(a|0)] = 3√(3) |a - x|/√(2π)∫_0^+∞e^-3(a - x)^2/2s/t^3/2 dt .

Notice that, unlike the case of BM, it is finite (in fact ρ(t) = t^3/3
∼ c· t^αα = 3 > 2 (see the observation after equation (<ref>)).
Calculating numerically the integral e.g.
for a = 1, we have obtained  E[τ_1(a|0)] ≈  3.594  , which is
not far from the estimate Ê[τ_1(a|0)] = 3.704  , obtained by Monte Carlo simulation.
By using (<ref>), one has:

    f_T_2(t)    = ∫_0^+∞3|a - x|/π (t + s)^3[∫_0^(t + s)^3/3e^-(a - x)^2/2u/u√((t + s)^3/3
    - u)du](t + s)^2s^23√(3)|a - x|/s^5/2e^-3(a - x)^2/2s^3/√(2π)ds
       = 9√(3)(a -
    x)^2/2π^2∫_0^+∞e^-3(a - x)^2/2s^3/(t + s)√(s)[∫_0^(t + s)^3/3e^-(a - x)^2/2u/u√((s
    + t)^3/3 - u)du]ds.


In the Figure <ref>, we report the probability density of T_2(x), obtained from (<ref>) by numerical computation, for
various values of the parameter a.




By using (<ref>), one gets the density of τ_2 (a|x):

    f_τ_ 2(t) = 3/π t∫_0^t^3/3|a - x|/√(2π)  s √(t^3/3 - s) e^-(a - x)^2/2s ds .

Calculating numerically the integral, we obtained the shape  of the density of τ_ 2(a|x) , for different values of the parameter a; the results are shown in the Figure <ref>.



As for the expectation of τ_2(a|0), one has
E[τ_ 2(a|0)] = - d /d λψ̃ ̃(̃ ̃λ̃ ̃)̃_λ = 0,
where  ψ̃(λ) is  the Laplace transform of τ_2(a|x), i.e.

    ψ̃(λ) =  ∫ _0 ^  + ∞ e^  - λ t  f_ τ _2 (t) dt = 3|a - x|/π√(2π)∫_0^+∞e^-λ t/t[∫_0^t^3/3e^-(a -
    x)^2/2s/s√(t^3/3 - s)ds]dt.

Thus:

    E[τ_ 2(a|x)] = 3|a - x|/π√(2π)∫_0^+∞[∫_0^t^3/3e^-(a -
    x)^2/2s/s√(t^3/3 - s) ds] dt .

For not too large values of a, Monte Carlo  simulation provides the estimate E[τ_2(a|0)], not  far from the
exact value, obtained by using (<ref>).






In the Figure <ref>, we report the comparison between the density, (<ref>), of τ_2(a|0) and the density, (<ref>),
of τ_ 1(a|0), for  a = 1 .
Notice that in both the Figures <ref> and <ref> the curves appear to be somewhat jagged, due to approximation in
numerical computation. This also happens in the next Figure 13 and Figure 15.


We conclude the example of B(t^3/3),  reporting the LTs of τ_1(a|0) and τ _2 (a|0). 

∙ LT of τ_1(a|x):


    ψ(λ) = E[e^- λτ_1(a|x) ] = ∫_0^+∞f_τ_1(t)e^-λ t dt =
    ∫_0^+∞3√(3)|a - x|/t^5/2  e^-λ t·e^-3(a - x)^2/2t^3/√(2π) dt;

∙ LT of τ_2(a|x):

    ψ̃(λ) = E[e^- λτ_2(a|x) ] = ∫_0^+∞f_τ_2(t)e^-λ t dt =3|a -
    x|/π√(2π)∫_0^+∞e^-λ t/t[∫_0^t^3/3e^-(a - x)^2/2s/s√(t^3/3 -
    s) ds] dt .





2) The Ornstein-Uhlenbeck (OU) process It is the solution of:

    dX(t) = -μ X(t)dt + σ dB(t) 
     X(0) = x

with μ,  σ positive constants. Explicitly, one has

    X(t) = e^-μ t(x + ∫_0^t σ e^μ s dB(s)).

Moreover, (see e.g. <cit.>) X(t) can be written as e^  - μ t  ( x + B( ρ (t))  ),
where ρ (t)= σ ^2 / 2 μ  (1 - e^ -2 μ t  ), namely in terms of time-changed BM.
Therefore, X(t) has normal distribution with mean xe^-μ t and variance σ^2/2μ(1 - e^-2μ t) .
We choose the time-varying barrier S(t) = a  e^-μ t, with x < a, and  we  reduce to the passage-times of BM through the constant
barrier a. In fact:

    τ_1(S|x) = inf{t > 0 : X(t) = S(t)} =inf{t > 0 :
    x+ B(ρ(t)) = a }

and

    τ_1(S|x) = ρ^-1( τ_1 ^B (a|x)),


where τ_1 ^B (a|x) is the first passage time of BM, through the barrier a.
Moreover,

    τ_2(S|x) = ρ^-1(τ_2 ^B (a|x)), … , τ_n(S|x) = ρ^-1(τ_n ^B (a|x) ),

which are the successive-passage times of BM through a.
Thus, by using (<ref>), we get the density of τ _1 (S|x):

    f_τ_ 1(t) = 2μ√(2μ)·|a - x|/σ(e^2μ t - 1)^3/2·1/√(2π)· e^-μ(a -
    x)^2/σ^2(e^2μ t - 1) + 2μ t .


In the Figure <ref> we report the density of τ_1(S|x), for x = 0, μ = 2, σ = 0.2 and various values
of a.




As for the expectation of τ_1(a|x), we have:

    E[τ_1(S|x)] = |a - x|·2μ√(μ)/σ√(π)·∫_0^+∞ t  e^-(x - a)^2·μ/σ^2·(e^2μ t - 1) + 2μ t  dt ;

calculating numerically the integral e.g. for x=0,  a=1,  μ = σ =1, one obtains
E[τ_1(S|x)] = 1.025,
while, Monte Carlo simulation provides the estimate
E[τ _1(S|x)] = 1.048. By using (<ref>), one has:

    f_T_2(t)    = (2μ)^5/2σ(a - x)^2/2π^2·∫_0^+∞e^-μ(a - x)^2/σ^2(e^2μ s - 1) + 2μ(t + 3s)/(e^2μ(t +
    s) - 1)(e^2μ s - 1)^3/2·
       ·[∫_0^σ^2/2μ(e^2μ(t + s) - 1)e^-(a -
    x)^2/2u/u√(σ^2/2μ(e^2μ(t + s)-1)-u)du]ds .







In the Figure <ref> we report the probability density of T_2(x), obtained from (<ref>), by calculating numerically the integral, for
x = 0, σ = 0,2, μ = 2 and various values of the parameter a. Notice that, for a=3 the curve appears to be overlapped
with the time axis.
By using (<ref>), we obtain

    f_τ_2(t) = 2μ|a - x|e^2μ t/π√(2π)·(e^2μ t - 1)·∫_0^σ^2/2μ(e^2μ t - 1)e^-(a -
    x)^2/2s/s√(σ^2/2μ(e^2μ t - 1) - s) ds.

Calculating numerically the integral, one gets
the density of τ_2(S|x); for x = 0,  σ = 0.2,   μ = 2 and different values of the parameter a,
the results are shown in the
Figure <ref>.



We summarize the LTs of τ_1(S|x) and τ_2(S|x) in the case of OU process and S(t)= ae^-μ t. 

∙ LT of τ_1(S|x):

    ψ(λ) = E[e^- λτ_1(S|x)] =
    ∫_0^+∞f_τ_1(t)e^-λ t dt = (2μ)^3/2·|a - x|/σ√(2π)∫_0^+∞e^2μ
    t - μ(a - x)^2/σ^2(e^2μ t - 1) - λ t/(e^2μ t - 1)^3/2 dt .


∙  LT of τ_2(S|x):

    ψ̃(λ) = E[e^-λτ_2(S|x) ] = ∫_0^+∞f_τ_2(t)e^-λ t dt


    =2μ|a - x|/π√(2π)∫_0^+∞e^2μ t - λ t/e^2μ t - 1[∫_0^σ^2/2μ(e^2μ t -
    1)e^-(a - x)^2/2s/s√(σ^2/2μ(e^2μ t - 1) - s) ds]dt .





As far the expectation of τ_2(S|x) is concerned, it is
not possible to obtain the exact value,
by taking minus the derivative with respect to λ  in (<ref>), that is:

    - d /d λψ̃(λ)_λ = 0= 2μ|a - x|/π√(2π)·∫_0^+∞t
    e^2μ t/e^2μ t - 1·[∫_0^σ^2/2μ(e^2μ t - 1)e^-(a - x)^2/2s/s√(σ^2/2μ(e^2μ t - 1)
    - s) ds] dt ,

because of the complexity of the integrals involved in the calculation, so it has to be estimated
by Monte Carlo simulation.
In the Figure 14, with regard to OU through the boundary S(t)= a e^  - μ t,  a=1, we report the comparison between the LT of τ_1(S|x) and 
that of τ_2(S|x),
for x = 0, σ = 0.2, μ = 2;
in the Figure <ref>, we report the comparison between the density of τ_1(S|x) and that 
of τ_2(S|x), for the same set of parameters.





§ CONCLUSIONS AND FINAL REMARKS

By using the results of <cit.> on the nth-passage time of Brownian motion through  a straight line, we studied the distribution
of the nth-passage time through a barrier a, of a diffusion process X(t), given by a space or time transformation of
Brownian motion B_t; precisely, we have considered the cases: X(t)= v^-1 (B_t + v(x)), where v(x) is an increasing function with v(0)=0,
and X(t)= B ( ρ (t)), where ρ (t)  is an increasing function with ρ (0) =0.
We also found the Laplace transform of the first and second-passage time of such
processes, and we have reported some explicit examples. Notice that the results of this paper  can be extended to diffusions which are more general than the process X(t)  here considered,
for instance to a process of the form

    X(t)= v^ -1 ( B(ρ (t)) + v (x) ) ,

where B (t) is BM, v(x) and ρ (t)  are regular enough, increasing functions with ρ (0) = v(0) =0; such a process
X(t) is obtained from BM combining a space transformation and a
time-change (see e.g. the discussion in <cit.>).
When ρ (t) = t,
the process X(t) is conjugated to BM, according to the
definition given in Section 3.1 (see <cit.>). The process X(t) given by
(<ref>) is indeed a weak solution of the SDE:

    dX(t)= - ρ ' (t) v” (X(t)) / 2 (v' (X(t)))^3  dt + √(ρ '(t))/ v' (X(t))  d B_t  ,

where  v'(x) and v”(x) denote first and second derivative of
v(x).  Provided that the deterministic function ρ (t) is  replaced
with a random function, the representation (<ref>) is also valid  for a
time homogeneous one-dimensional diffusion driven by the SDE

    dX(t) = μ (X(t)) dt + σ (X(t)) d B_t ,  X(0) = x ,

where the drift ( μ ) and diffusion coefficients ( σ ^2 ) satisfy the usual conditions
(see e.g. <cit.>)
for existence and uniqueness
of the solution of (<ref>). In fact,
let w(x) be the scale function associated to
the diffusion X(t) driven by the SDE (<ref>), that is,
the  solution of L w(x) =0,  w(0) =0,  w'(0) =1 ,
where L is the infinitesimal generator of X given by
L h = 1/2σ ^2 (x) d^2 h /d x^2  + μ (x) d h /d x.
As easily seen,  if the integral ∫ _0 ^t  2 μ (z)/σ ^2 (z) dz converges, the scale function is explicitly given by

    w(x)= ∫ _ 0 ^x exp ( - ∫ _0  ^t  2 μ (z)/σ ^2 (z) dz  ) dt .

If ζ (t) := w(X(t)), by Itô's formula one
obtains

    ζ (t) = w ( x) + ∫ _0 ^t w '( w ^-1 (ζ (s)) ) σ ( w ^-1 (ζ  (s))) d B_s   ,

that is, the process ζ (t) is a local martingale, whose
quadratic variation is

    ρ (t) ≐⟨ζ⟩ _t = ∫ _0 ^ t [ w' (X(s)) σ (X(s)) ] ^ 2 ds,  t ≥ 0 .

The (random) function ρ(t) is differentiable, increasing, and
ρ (0)=0. If ρ ( + ∞) = + ∞,
by the Dambis, Dubins-Schwarz theorem (see e.g. <cit.>) one gets
that there exists a BM B such that ζ (t) = B ( ρ (t)) + w ( η ).
Thus, since w is invertible, one obtains the representation (<ref>) with w in place of u. Notice, however, that the successive-passage time problem for the process X(t) given by (<ref>) can be
addressed as in the  case when ρ (t) is a deterministic function, only if
the distribution of the random process ρ (t) is explicitly known.




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