First-exit problems for integrated diffusion processes with

state-dependent jumps

Department of Mathematics and Industrial Engineering

Polytechnique Montréal

2500, chemin de Polytechnique, Montréal (Québec) H3T 1J4

CANADA

https://www.polymtl.ca/expertises/en/lefebvre-mario

Abstract: Let dX(t) = −Y(t)dt, where Y(t)is a one-dimensional diffusion process. First-exit problems from

C∈R2are studied for the degenerate two-dimensional diffusion process (X(t), Y (t)) when the process leaves C

not later than after a random time having an exponential distribution. When Y(t)is a standard Brownian motion,

the Laplace transform of the moment-generating function Mof the first-exit time is computed explicitly, as well

as the Laplace transforms of the mean exit time mand the probability pof leaving Cthrough a given part of its

boundary. When Y(t)is a geometric Brownian motion, the functions M,mand pare obtained by making use

of the method of similarity solutions to solve the various partial differential equations, subject to the appropriate

boundary conditions.

Key-Words: Kolmogorov backward equation, Brownian motion, geometric Brownian motion, method of

similarity solutions.

1 Introduction

We consider degenerate two-dimensional diffusion

processes (X(t), Y (t)) defined by

dX(t) = −Y(t)dt, (1)

dY(t) = f[Y(t)]dt+{v[Y(t)]}1/2 dB(t),(2)

where {B(t), t ≥0}is a standard Brownian motion

and the functions fand vare such that {Y(t), t ≥0}

is a diffusion process. The process {X(t), t ≥0}

(multiplied by −1) is known as an integrated diffusion

process, because we can write that

X(t) = X(0) −Zt

Y(s)ds. (3)

Numerous papers have been written on first-exit

problems for integrated diffusion processes. For

the case when {Y(t), t ≥0}is a Wiener pro-

cess, see McKean [12], Goldman [2], Gor’kov [3],

Lachal [5] and Lefebvre [8]. Moreover, Lefebvre [7],

Makasu [11], Metzler [13], Caravelli et al. [1] and

Levy [10] considered the case of integrated geomet-

ric Brownian motions. Finally, Lefebvre [6] and

Hesse [4] studied problems for integrated Ornstein-

Uhlenbeck processes.

Next, let (X(0), Y (0)) = (x, y)∈C⊂R2and

define the first-exit time from C

T(x, y) = inf{t > 0 : (X(t), Y (t)) /∈C}.(4)

Notice that if Y(t)is always positive in C, then X(t)

will be strictly decreasing with time. Therefore, this

type of process can serve as model in applications

where the variable of interest cannot increase, for in-

stance when X(t)represents the remaining lifetime

of a certain device.

Assume that α > 0. The function

M(x, y) := Ehe−αT (x,y)i(5)

is the moment-generating function of the random

variable T(x, y). It satisfies the Kolmogorov back-

ward equation (see Lefebvre [9], for a generalization

of this result)

2v(y)Myy +f(y)My−y Mx=αM (6)

for (x, y)∈C, where Myy := ∂2

∂y2M, etc. Further-

more, since T(x, y) = 0 if (x, y) /∈C, we have the

boundary condition

M(x, y;α) = 1 for (x, y) /∈C. (7)

Now, suppose that at a random time τhav-

ing an exponential distribution with parameter λ, if

(X(t), Y (t)) is still inside Ca random quantity Zis

added to the value of Y(t)at that time instant, so that

MARIO LEFEBVRE

Received: April 25, 2022. Revised: October 25, 2022. Accepted: November 28, 2022. Published: December 31, 2022.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.98

Mario Lefebvre

E-ISSN: 2224-2880

864

Volume 21, 2022

the process (X(t), Y (t)) will exit the continuation re-

gion Cimmediately. We say that the process is killed

not later than at time τ.

We can prove the following proposition.

Proposition 1.1. The function M(x, y)satisfies the

partial differential equation (PDE)

αM(x, y) = 1

2v(y)Myy(x, y) + f(y)My(x, y)

−y Mx(x, y) + λ[1 −M(x, y)].(8)

The equation is valid for (x, y)∈Cand is subject to

the boundary condition (7).

We also have the following corollaries.

Corollary 1.1. If it exists, the function m(x, y) :=

E[T(x, y)] satisfies the PDE

−1 = 1

2v(y)myy(x, y) + f(y)my(x, y)

−y mx(x, y)−λm(x, y)(9)

for (x, y)∈C. The boundary condition is

m(x, y) = 0 for (x, y) /∈C.(10)

Corollary 1.2. Let

p(x, y) := P[(X(T), Y (T)) ∈∂C0],(11)

where ∂C0is a subset of the boundary ∂C of C. For

(x, y)∈C, the probability p(x, y)is a solution of the

PDE

0 = 1

2v(y)pyy(x, y) + f(y)py(x, y)

−y px(x, y) + λ[p0−p(x, y)],(12)

where p0:= P[(X(τ), Y (τ)) ∈∂C0]. Furthermore,

the boundary condition is

p(x, y) = 1if (x, y)∈∂C0,

0if (x, y)∈∂D,(13)

where ∂D := ∂C \∂C0.

In the next section, we will compute the Laplace

transform of the functions M(x, y),m(x, y)and

p(x, y)when {Y(t), t ≥0}is a standard Brown-

ian motion and T(x, y)is the first time that the two-

dimensional process ((X(t), Y (t)) leaves a rectangle

located in the first quadrant. In Section 3, the set

Cwill be the region between two straight lines and

{Y(t), t ≥0}will be a particular geometric Brown-

ian motion. With the help of the method of similarity

solutions, we will be able to get the exact solutions to

the PDE’s (8), (9) and (12). We will end this paper

with a few remarks in Section 4.

2 Integrated Brownian motion

In this section, we assume that v(y)≡1and f(y)≡

0, so that {Y(t), t ≥0}is a standard Brownian mo-

tion, and we define

T1(x, y)=inf{t > 0 : X(t) = 0 or Y(t) = aor b},

(14)

where x > 0and 0≤a<y<b. Moreover, the

quantity Zadded to Y(t)at time τis

Z=b−Y(τ)with probability p1∈(0,1),

a−Y(τ)with probability 1−p1.

(15)

Because Y(t)is always positive in Cand dX(t) =

−Y(t)dt,X(t)will be strictly decreasing with time.

Moreover, Cis actually a finite rectangle located in

the first quadrant.

Solving the PDE (8) is not an easy task. We can

however obtain the Laplace transform of the function

M(x, y)explicitly. Indeed, let

L(y) := Z∞

e−βx M(x, y)dx, (16)

where β > 0. Since M(x, y)∈(0,1) for (x, y)∈C

and M(0, y) = 1, we have

Z∞

e−βx Mx(x, y)dx=e−βx M(x, y)

∞

+β L

=−1 + β L. (17)

It follows, taking the Laplace transform of each side

of Eq. (8), that

αL(y) = 1

2L00(y)−y[−1 + β L(y)]

+λ[(1/β)−L(y)].(18)

The above ordinary differential equation (ODE) is

subject to the boundary conditions

L(a) = L(b) = 1/β. (19)

Making use of the mathematical software program

Maple, we find that the general solution of Eq. (18)

can be written as follows:

L(y) = c1Ai(η) + c2Bi(η)

+22/3 π

β4/3 Ai(η)ZBi(η)(β y +λ)dy

−Bi(η)ZAi(η)(β y +λ)dy,(20)

where Ai and Bi are Airy functions and

η:= 21/3

β2/3 (β y +λ+α).(21)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.98

Mario Lefebvre

E-ISSN: 2224-2880

865

Volume 21, 2022

The arbitrary constants c1and c2can be determined

from the boundary conditions (19) for any choice of

aand b. The expression obtained is rather involved.

Therefore, inverting the Laplace transform to obtain

the function M(x, y)would be really difficult.

Next, let

L1(y) := Z∞

e−βx m(x, y)dx. (22)

Proceeding as above, we find that Eq. (9) becomes the

ODE

−1

β=1

2L00

1(y)−(y β +λ)L1(y),(23)

whose general solution is

L1(y) = c1Ai(ξ) + c2Bi(ξ)(24)

+22/3 π

β4/3 Ai(ξ)ZBi(ξ)dy−Bi(ξ)ZAi(ξ)dy,

where

ξ:= 21/3

β2/3 (β y +α).(25)

The constants c1and c2must be chosen so that

L1(a) = L1(b) = 0.

Finally, let

p1(x, y) := P[X(T) = 0].(26)

Since p0=P[X(τ) = 0] = 0, we may write that the

function

L2(y) := Z∞

e−βx p1(x, y)dx(27)

satisfies the ODE

0 = 1

2L00

2(y)−(y β +λ)L2(y) + y. (28)

Indeed, we calculate

Z∞

e−βx ∂

∂xp1(x, y)dx=e−βx p1

∞

+β L2

= [0 −p1(0, y)] + β L2

=−1 + β L2.(29)

Maple gives us the following expression for the gen-

eral solution of Eq. (28):

L2(y) = c1Ai(ξ) + c2Bi(ξ)(30)

+22/3 π

β1/3 Ai(ξ)ZBi(ξ)dy−Bi(ξ)ZAi(ξ)dy.

Notice that the functions L1(y)and L2(y)are almost

the same. Moreover, the constants c1and c2are also

such that L2(a) = L2(b) = 0.

In the next section, we will be able to obtain the

exact expressions for the functions of interest.

3 Integrated geometric Brownian

motion

Suppose that the diffusion process {Y(t), t ≥0}is

defined by

dY(t) = k Y (t)dt+Y(t)dB(t).(31)

That is, {Y(t), t ≥0}is a geometric Brownian

motion with infinitesimal mean k y and infinitesimal

variance y2. This process can be expressed as the ex-

ponential of a Wiener process with infinitesimal pa-

rameters k−1

2and 1. Therefore, if Y(0) >0, the

process will remain positive for any t > 0, and X(t)

will be strictly decreasing with t.

Let

T2(x, y) := inf{t > 0 : X(t)/Y(t) = k1or k2},

(32)

where 0< k1< x/y < k2. Thus, the set Cis the

region between two straight lines going through the

origin.

Next, assume that

Z=(1

k1X(τ)−Y(τ)with probability p1,

k2X(τ)−Y(τ)with probability 1−p1,

(33)

so that the process (X(t), Y (t)) will indeed leave the

continuation region Cnot later than at time τ.

The PDE given in (8) becomes

αM =1

2y2Myy +k y My−y Mx+λ[1 −M(x, y)].

(34)

It is subject to the boundary conditions M(x, y)=1

if x/y=k1or k2.

Now, based on Eq. (34) and the boundary con-

ditions, we will look for a solution of the form

M(x, y) = N(r), where r:= x/yis called a sim-

ilarity variable. We find that Eq. (8) is transformed

into the ODE

2r2N00 +(r−k r −1)N0−(α+λ)N+λ= 0.(35)

The boundary conditions become N(k1) = N(k2) =

The general solution of Eq. (35) is

N(r)= rk−ζ+1

2e−2/rc1M1 + ζ

2+k, 1 + ζ, 2

r

+c2U1 + ζ

2+k, 1 + ζ, 2

r+λ

α+λ,(36)

where

ζ:= p4(k2−k) + 8(α+λ)+1,(37)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.98

Mario Lefebvre

E-ISSN: 2224-2880

866

Volume 21, 2022

M(·,·,·)and U(·,·,·)are Kummer functions, and the

constants c1and c2are determined from the boundary

conditions N(k1) = N(k2) = 1. In the special case

when k= 1, the solution can be expressed in terms

of the Bessel functions Iν(·)and Kν(·)as follows:

N(r) = c1[(γ+ 1)r+ 2]Iγ

2(1/r)+2Iγ

2+1(1/r)

+c2[(γ+ 1)r+ 2]Kγ

2(1/r)−2Kγ

2+1(1/r)

×e−1/r

√r+λ

α+λ,(38)

where

γ:= p8(α+λ)+1.(39)

Next, we assume that the function m2(x, y) :=

E[T2(x, y)] can be written as N1(r). Proceeding as

above, we find that the function N1(r)satisfies the

ODE

−1 = 1

2r2N00

1(r)+(r−k r−1)N0

1(r)−(α+λ)N1(r),

(40)

subject to N1(k1) = N1(k2)=0. The general solu-

tion of Eq. (40) is

N1(r) = c1M1 + ζ0

2+k, 1 + ζ0,2

r

+c2U1 + ζ0

2+k, 1 + ζ0,2

r

×rk−ζ0+1

2e−2/r+1

λ,(41)

where

ζ0:= p4(k2−k)+8λ+ 1.(42)

This solution can also be expressed in terms of Bessel

functions when k= 1.

Finally, we define

p2(x, y) = P[X(T2)/Y(T2) = k1].(43)

We assume that p2(x, y) = N2(r). To obtain the

function N2(r), we must find the solution of the ODE

0 = 1

2r2N00

2(r)+(r−k r −1)N0

2(r)−λN2(r)+λp1

(44)

that is such that N2(k1) = 1 and N2(k2) = 0. We

find that the general solution of the above equation is

N2(r) = c1M1 + ζ0

2+k, 1 + ζ0,2

r

+c2U1 + ζ0

2+k, 1 + ζ0,2

r

×rk−ζ0+1

2e−2/r+p1.(45)

As in the previous cases, the function N2(r)can be

expressed in terms of Bessel functions when k= 1.

4 Conclusion

First-exit problems for diffusion processes have ap-

plications in many areas. To solve such problems,

one usually needs to find the solution of a differential

equation that satisfies certain boundary conditions. In

two or more dimensions, the equation to be solved is

a PDE. Therefore, the problem is difficult.

In this paper, we have added the constraint that

the two-dimensional process leaves the continuation

region at the latest at a random time that follows an

exponential distribution, which further increases the

difficulty of the problems considered.

In Section 2, we treated the case where the diffu-

sion process {Y(t), t ≥0}is a standard Brownian

motion and the continuation region Cis a rectangle

located in the first quadrant. We have succeeded in

obtaining the Laplace transform of the three functions

of interest.

In Section 3, {Y(t), t ≥0}was a geometric Brow-

nian motion and the set Cwas the region between two

straight lines that intersect at the origin. In this case,

we obtained, using the method of similarity solutions,

the exact expressions for the functions of interest.

As a continuation of this work, we could try to

solve this type of problem when the jumps are contin-

uous rather than discrete random variables. We could

also try to solve stochastic optimal control problems

defined in terms of the processes studied.

Acknowledgements

This research was supported by the Natural Sci-

ences and Engineering Research Council of Canada.

References:

[1] Caravelli, F., Mansour, T., Sindoni, L. and Sev-

erini, S., On moments of the integrated expo-

nential Brownian motion, The European Phys-

ical Journal Plus, Vol. 131, 2016, Article 245.

https://doi.org/10.1140/epjp/i2016-16245-9

[2] Goldman, M., On the first passage of the inte-

grated Wiener process, Annals of Mathematical

Statistics, Vol. 42, No. 6, 1971, pp. 2150–2155.

https://doi.org/10.1214/aoms/1177693084

[3] Gor’kov, Yu. P., A formula for the solution of a

certain boundary value problem for the station-

ary equation of Brownian motion, Soviet Math-

ematics. Doklady, Vol. 16, 1975, pp. 904–908.

[4] Hesse, C. H., The one-sided barrier prob-

lem for an integrated Ornstein-Uhlenbeck pro-

cess, Communications in Statistics. Stochastic

Models, Vol. 7, No. 3, 1991, pp. 447–480.

https://doi.org/10.1080/15326349108807200

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.98

Mario Lefebvre

E-ISSN: 2224-2880

867

Volume 21, 2022

[5] Lachal, A., L’intégrale du mouvement

brownien, Journal of Applied Proba-

bility, Vol. 30, No. 1, 1993, pp. 17–27.

https://doi.org/10.2307/3214618

[6] Lefebvre, M., Moment generating function of

a first hitting place for the integrated Ornstein-

Uhlenbeck process, Stochastic Processes

and their Applications, Vol. 32, No. 2, 1989,

pp. 281–287. https://doi.org/10.1016/0304-

4149(89)90080-X

[7] Lefebvre, M., First hitting time and place for the

integrated geometric Brownian motion, Interna-

tional Journal of Differential Equations and Ap-

plications, Vol. 9, No. 4, 2004, pp. 365–374.

[8] Lefebvre, M., Moments of first-passage places

and related results for the integrated Brownian

motion, ROMAI Journal, Vol. 2, No. 2, 2006,

pp. 101–108.

[9] Lefebvre, M., A first-passage problem for expo-

nential integrated diffusion processes, Journal

of Stochastic Analysis, Vol. 3, No. 3, 2022, Arti-

cle 2. https://doi.org/10.31390/josa.3.3.02

[10] Levy, E., On the moments of the inte-

grated geometric Brownian motion, Jour-

nal of Computational and Applied Math-

ematics, Vol. 342, 2018, pp. 263–273.

https://doi.org/10.1016/j.cam.2018.04.005

[11] Makasu, C., Exit probability for an

integrated geometric Brownian mo-

tion, Statistics & Probability Letters,

Vol. 79, No. 11, 2009, pp. 1363–1365.

https://doi.org/10.1016/j.spl.2009.02.009

[12] McKean, H. P., A winding problem for

a resonator driven by a white noise,

Journal of Mathematics of Kyoto Uni-

versity, Vol. 2, 1963, pp. 227–235.

https://doi.org/10.1215/kjm/1250524936

[13] Metzler, A., The Laplace transform of hit-

ting times of integrated geometric Brown-

ian motion, Journal of Applied Probabil-

ity, Vol. 50, No. 1, 2013, pp. 295–299.

https://doi.org/10.1239/jap/1363784440

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/li-

censes/by/4.0/deed.en_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.98

Mario Lefebvre

E-ISSN: 2224-2880

868

Volume 21, 2022