Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2012, Article ID 971212, 15 pages
doi:10.1155/2012/971212
Research Article
The First Passage Time and the Dividend Value
Function for One-Dimensional Diffusion Processes
between Two Reflecting Barriers
Chuancun Yin1 and Huiqing Wang1, 2
1
2
School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
School of Economics and Management, Southeast University, Nanjing 211189, China
Correspondence should be addressed to Chuancun Yin, ccyin@mail.qfnu.edu.cn
Received 26 July 2012; Accepted 24 September 2012
Academic Editor: Enzo Orsingher
Copyright q 2012 C. Yin and H. Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the general one-dimensional time-homogeneous regular diffusion process between
two reflecting barriers. An approach based on the Itô formula with corresponding boundary
conditions allows us to derive the differential equations with boundary conditions for the Laplace
transform of the first passage time and the value function. As examples, the explicit solutions of
them for several popular diffusions are obtained. In addition, some applications to risk theory are
considered.
1. Introduction and the Model
Diffusion processes with one or two barriers appear in many applications in economics,
finance, queueing, mathematical biology, and electrical engineering. Among queueing system
applications, reflected Ornstein-Uhlenbeck and reflected affine processes have been studied
as approximations of queueing systems with reneging or balking 1, 2. Motivated by Ward
and Glynn’s one-sided problem, Bo et al. 3 considered a reflected Ornstein-Uhlenbeck
process with two-sided barriers. In this paper, we consider the expectations of some random
variables involving the first passage time and local times for the general one-dimensional
diffusion processes between two reflecting barriers.
Let X {Xt , t ≥ 0} be a one-dimensional time-homogeneous reflected diffusion
process with barriers a and b, which is defined by the following stochastic differential
equation:
dXt μXt dt
σXt dBt
X0 x ∈ a, b,
dLt − dUt ,
1.1
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International Journal of Stochastic Analysis
where Bt is a Brownian motion in R, L {Lt , t ≥ 0} and U {Ut , t ≥ 0} are the regulators
of point a and b, respectively. Further, the processes L and U are uniquely determined by the
following properties see, e.g., 4:
1 both t → Lt and t → Ut are continuous nondecreasing processes with L0 U0 0,
t
2 L and U increase only when X a and X b, respectively, that is, 0 I{Xs a} dLs Lt
t
and 0 I{Xs b} dUs Ut , for t ≥ 0.
It is well known that under certain mild regularity conditions on the coefficients μx
and σx, the SDE 1.1 has a unique strong solution for each starting point see, e.g., 5.
The solution Xt is a time-homogeneous strong Markov process with infinitesimal generator
Afx 1 2
σ xf x
2
μxf x,
x ∈ a, b,
1.2
acting on functions on a, b subject to boundary conditions: f a f b 0.
Define the first passage time
τy inf t ≥ 0 : Xt y ,
1.3
where τy ∞ if Xt never reaches y.
For λ > 0, η > 0, θ > 0, we consider the Laplace transform ϕ, and the value functions
ψ, ψ1 , and ψ2 on x ∈ a, b:
ϕx Ex e−λτy ,
1.4
∞
∞
−λt
ψx Ex η
e dUt − θ
e−λt dLt ,
0
ψ1 x Ex
∞
1.5
0
e−λt dUt ,
1.6
e−λt dLt .
1.7
0
ψ2 x Ex
∞
0
The rest of the paper is organized as follows. Section 2 studies the Laplace transform
of the first passage time. Section 3 deals with the value function. Some applications in risk
theory are considered in Section 4.
2. Laplace Transform
Bo et al. 3 consider the Laplace transform Ex e−λτy for a reflected Ornstein-Uhlenbeck
process with two-sided barriers. In this section we consider the Laplace transform of the first
passage time for the general reflected diffusion process X defined by 1.1.
International Journal of Stochastic Analysis
3
Theorem 2.1. Let x ∈ a, b, λ > 0, and assume that f1 y, f2 y satisfy the following equations,
respectively:
Af1 y λf1 y ,
y ∈ a, b,
f1 a 0,
Af2 y λf2 y ,
2.1
y ∈ a, b,
f2 b 0.
0 for y ∈ x, b and f2 y /
0 for y ∈ a, x, then
If f1 y /
f1 x
Ex e−λτy ,
f1 y
y ∈ x, b,
f2 x
,
Ex e−λτy f2 y
y ∈ a, x.
2.2
Proof. Applying the Itô formula for semimartingales to ht, Xt e−λt fXt with f ∈
Cb2 a, b we obtain
t
ht, Xt h0, X0 0
t
t
∂h
hs, Xs ds
∂s
0
∂h
hs, Xs dXs
∂x
2
∂h
hs, Xs dXs , Xs 2
0 ∂x
t
t
fX0 −λe−λs fXs ds
e−λs f Xs dXs
1
2
1
2
t
fx
0
0
0
2.3
e−λs f Xs σ 2 Xs ds
t
e−λs A − λfXs ds
0
f a
t
e−λs dLs − f b
0
t
t
e−λs f Xs dBs
0
e−λs dUs .
0
Since τy < ∞ is a stopping time and x ∈ a, b, it follows from the optional sampling theorem
that
Ex e−λτy f Xτy
fx
τy
Ex
e−λs A − λfXs ds
0
f aEx
τy
0
e−λs dLs
− f bEx
τy
0
e−λs dUs .
2.4
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International Journal of Stochastic Analysis
By the definitions of τy , U, and L we have
τy
Ex
e−λs dLs 0,
τy
y ∈ a, x;
Ex
0
e−λs dUs 0,
y ∈ x, b;
X τy y.
2.5
0
Substituting them into 2.4 one gets
Ex e−λτy f y fx
τy
Ex
e−λs A − λfXs ds
0
f aEx
τy
e−λs dLs ,
y ∈ x, b.
0
Ex e−λτy f y fx
2.6
τy
Ex
e
−λs
A − λfXs ds
0
− f bEx
τy
e−λs dUs ,
y ∈ a, x.
0
The result follows.
Remark 2.2. Although neither f1 · nor f2 · in the Theorem 2.1 is unique, but each of their
ratios is unique.
As illustrations of Theorem 2.1, we consider some examples.
Example 2.3. Bessel process: dXt d − 1/2Xt dt dBt , where d > 1 is a real number.
We consider the differential equation 1/2ψ x d − 1/2xψ x λψx, λ > 0. It
is well known that the increasing and decreasing solutions are, respectively:
ψ x x−ν Iν
2λx ,
ψ− x x−ν Kν
2λx ,
2.7
where ν d − 2/2 and Iν and Kν are the usual modified Bessel functions.
Then, we can give f1 x, f2 x as follows:
f1 x C1 ψ x
C2 ψ− x,
f2 x C3 ψ x
C4 ψ− x,
2.8
where the constants C1 , C2 and C3 , C4 can be derived from f1 a 0 and f2 b 0,
respectively. We can obtain their ratios, respectively:
√
√
√
2λaKν
2λa − νKν 2λa
C1
M
− √
,
√
√
C2
2λaIν
2λa − νIν 2λa
√
√
√
2λbKν
2λb − νKν 2λb
C3
− √
.
N
√
√
C4
2λbIν
2λb − νIν 2λb
2.9
International Journal of Stochastic Analysis
5
Substituting them into 2.2, we get
Ex e−λτy Ex e−λτy √
Mx−ν Iν
2λx
x−ν Kν
2λy
y−ν Kν
2λx
x−ν Kν
2λy
y−ν Kν
√
My−ν Iν
√
Nx−ν Iν
√
Ny−ν Iν
√
2λx
,
√
2λy
y ∈ x, b,
2.10
√
2λx
,
√
2λy
y ∈ a, x.
Example 2.4. The Ornstein-Uhlenbeck process 6 is as follows:
dXt vk − Xt dt
σdBt ,
v, σ > 0, k ∈ a, b.
2.11
In mathematical finance, the Ornstein-Uhlenbeck process above is known as Vasicek
model for the short-term interest rate process 7. We consider the differential equation
1/2σ 2 ψ x vk − xψ x λψx.
In the case k 0, σ 1, the two independent solutions to 1/2ψ x−vxψ x λψx
are
√
√
2
ψ x H−λ/v − vx 2−λ/2v e1/2vx D−λ/v − 2vx ,
ψ− x H−λ/v
√
vx 2−λ/2v e1/2vx Dλ/v
2
√
2vx ,
2.12
where Hν · and Dν · are, respectively, the Hermite and parabolic cylinder functions 8.
Then, as the way used in Example 2.3, we obtain the ratios of the constants C1 , C2 and C3 , C4 ,
respectively:
√
√
√
2vDλ/v
2va
vaDλ/v 2va
C1
M
−
,
√
√
√
C2
vaD−λ/v 2va − 2vD−λ/v
2va
N
vbDλ/v
C3
−
C4
vbD
−λ/v
√
√
√
2vb
2vDλ/v
2vb
.
√
√
√
2vb
2vb − 2vD−λ/v
2.13
Substituting them into 2.2, we get
√
2
Me1/2vx D−λ/v − 2vx
Ex e−λτy √
2
Me1/2vy D−λ/v − 2vy
√
2
Ne1/2vx D−λ/v − 2vx
Ex e−λτy √
2
Ne1/2vy D−λ/v − 2vy
e1/2vx Dλ/v
2
e1/2vy Dλ/v
2
e1/2vx Dλ/v
2
e1/2vy Dλ/v
2
√
2vx
,
√
2vy
y ∈ x, b,
2.14
√
2vx
,
√
2vy
y ∈ a, x.
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International Journal of Stochastic Analysis
For the general k and σ, the two independent solutions are, respectively
√
√
v
2v
−λ/2v 1/2v/σ 2 x−k2
ψ x H−λ/v −
e
D−λ/v −
x − k 2
x − k ,
σ
σ
√
√
v
2v
−λ/2v 1/2v/σ 2 x−k2
ψ− x H−λ/v
e
D−λ/v
x − k 2
x − k .
σ
σ
2.15
Then, as the way used in Example 2.3, we obtain the ratios of the constants C1 , C2 and C3 , C4 ,
respectively
√
√
√
√
va − kD−λ/v
2v/σ a − k
2v/σ a − k
2σD−λ/v
C1
−√
,
M
√
√
√
C2
− 2v/σ a − k
va − kD−λ/v − 2v/σ a − k − 2σD−λ/v
√
√
√
√
vb − kD−λ/v
2v/σ b − k
2σD−λ/v
2v/σ b − k
C3
−√
.
N
√
√
√
C4
− 2v/σ b − k
vb − kD−λ/v − 2v/σ b − k − 2σD−λ/v
2.16
Substituting them into 2.2, we get
√
hx MD−λ/v − 2v/σ x − k
Ex e−λτy √
h y MD−λ/v − 2v/σ y − k
√
hx ND−λ/v − 2v/σ x − k
−λτy
Ex e
√
h y ND−λ/v − 2v/σ y − k
D−λ/v
D−λ/v
D−λ/v
D−λ/v
√
2v/σ x − k
√
2v/σ y − k
,
y ∈ x, b,
,
y ∈ a, x,
√
2v/σ x − k
√
2v/σ y − k
2.17
where hx e1/2v/σ
2
x−k2
.
Remark 2.5. If we take k 0, σ 1, a 0 and substitute the series forms of Hν · and Dν ·
into the above result, then it is the same as Bo et al. 3.
Example 2.6. The square root process of Cox et al. 9:
dXt vk − Xt dt
σ
Xt dBt ,
v, σ > 0, k ∈ a, b.
2.18
Now consider the differential equation
1 2 σ xψ x
2
vk − vxψ x λψx,
λ > 0.
2.19
International Journal of Stochastic Analysis
7
If 2v/σ 2 k is not an integer, the two linear independent solutions are
ψ x M
λ 2v 2v
, 2 k, 2 x ,
v σ
σ
ψ− x U
λ 2v 2v
, 2 k, 2 x ,
v σ
σ
2.20
where M and U are the confluent hypergeometric functions of the first and second kinds,
respectively. Then, as the way used in Example 2.3, we obtain the ratios of the constants C1 ,
C2 and C3 , C4 , respectively:
M1 U λ/v, 2v/σ 2 k, 2v/σ 2 a
C1
,
− C2
M λ/v, 2v/σ 2 k, 2v/σ 2 a
2.21
U λ/v, 2v/σ 2 k, 2v/σ 2 b
C3
.
− N
C4
M λ/v, 2v/σ 2 k, 2v/σ 2 b
Substituting them into 2.2, we get
M1 M λ/v, 2v/σ 2 k, 2v/σ 2 x
Ex e−λτy M1 M λ/v, 2v/σ 2 k, 2v/σ 2 y
NM λ/v, 2v/σ 2 k, 2v/σ 2 x
Ex e−λτy NM λ/v, 2v/σ 2 k, 2v/σ 2 y
U λ/v, 2v/σ 2 k, 2v/σ 2 x
,
U λ/v, 2v/σ 2 k, 2v/σ 2 y
U λ/v, 2v/σ 2 k, 2v/σ 2 x
U λ/v, 2v/σ 2 k, 2v/σ 2 y
,
y ∈ x, b.
y ∈ a, x.
2.22
Example 2.7. The Gompertz Brownian motion process 10 is as follows:
dXt vXtlnk − lnXt dt
σXt dBt ,
v, σ > 0, k ∈ a, b.
2.23
Now consider the differential equation
1 2 2 σ x ψ x
2
vxln k − ln xψ x λψx,
λ > 0.
2.24
The increasing and decreasing solutions are, respectively:
⎛
λ 1 v
ψ x M⎝ , , 2
2v 2 σ
⎛
ψ− x U⎝
λ 1 v
, ,
2v 2 σ 2
x
ln
k
ln
x
k
σ2
2v
2 ⎞
⎠,
2 ⎞
σ
⎠,
2v
2
2.25
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International Journal of Stochastic Analysis
where M and U, as in Example 2.6, are the first and second Kummer’s functions, respectively.
Then, as the way used in Example 2.3, we obtain the ratios of the constants C1 , C2 and C3 , C4 ,
respectively:
2
U λ/2v, 1/2, v/σ 2 lna/k σ 2 /2v
C1
M1 −
,
C2
M λ/2v, 1/2, v/σ 2 lna/k σ 2 /2v2
N
U λ/2v, 1/2, v/σ 2
lnb/k
C3
−
C4
M λ/2v, 1/2, v/σ 2 lnb/k
σ 2 /2v
2.26
2
σ 2 /2v2
.
Substituting them into 2.2, we get
2
M1 M λ/2v, 1/2, v/σ 2 lnx/k σ 2 /2v
−λτy
Ex e
M1 M λ/2v, 1/2, v/σ 2 lnx/k σ 2 /2v2
2
NM λ/2v, 1/2, v/σ 2 lnx/k σ 2 /2v
−λτy
Ex e
NM λ/2v, 1/2, v/σ 2 lnx/k σ 2 /2v2
where A denotes λ/2v, 1/2, v/σ 2 lnx/k
UA
y ∈ x, b,
,
UA
2.27
UA
,
UA
y ∈ a, x,
2
σ 2 /2v .
Remark 2.8. For a certain choice of parameters for a and b in Examples 2.3–2.7, we get the
Laplace transform of the first passage time of one-dimensional diffusion with one-sided
barrier. For example, letting a → −∞ or b → ∞ in Example 2.4, one gets the Laplace
transform of the first passage time of the Ornstein-Uhlenbeck process with one-sided barrier;
see Nobile et al. 11, Ricciardi and Sato 12, Alili et al. 13, and Ditlevsen 6.
3. The Value Function
In this section we study the value functions 1.5–1.7. Using Itô’s formula, we derive
differential equation with boundary conditions for ψ.
Theorem 3.1. The function ψ defined by 1.5 satisfies the differential equation
1 2
σ xψ x
2
μxψ x λψx,
with the boundary conditions ψ a θ, ψ b η.
a < x < b,
3.1
International Journal of Stochastic Analysis
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Proof. Applying the Itô’s formula for semimartingales to ht, Xt e−λt ψXt with ψ ∈
Cb2 a, b we obtain
t
ht, Xt h0, X0 0
t
t
∂h
hs, Xs ds
∂s
0
∂h
hs, Xs dXs
∂x
∂2 h
hs, Xs dXs , Xs 2
0 ∂x
t
t
ψX0 −λe−λs ψXs ds
e−λs ψ Xs dXs
1
2
t
1
2
0
ψx
0
0
e−λs ψ Xs σ 2 Xs ds
t
e−λs A − λψXs ds
0
3.2
ψ a
t
e
−λs
dLs − ψ b
t
0
t
e−λs ψ Xs dBs
0
e−λs dUs ,
0
where we have used that ψ Xt dLt ψ adLt and ψ Xt dUt ψ bdUt . From 3.2 we
have
Ex e−λt ψXt − ψx Ex
t
e−λs A − λψXs ds
0
Ex ψ a
t
e
−λs
dLs − ψ b
0
t
e
−λs
3.3
dUs .
0
Let f be a solution of
Afx λfx,
x ∈ a, b,
f a 0,
3.4
f b 1.
In place of 3.3, we have
Ex e−λt fXt − fx −
t
e−λs dUs .
3.5
0
Letting t → ∞, we get
∞
0
e−λs dUs fx < ∞.
3.6
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International Journal of Stochastic Analysis
Likewise
∞
e−λs dLs < ∞.
3.7
0
Letting t → ∞ in 3.3 and noting 3.6 and 3.7, we get the desired result.
∞
Corollary 3.2. The function ψ1 x Ex 0 e−λt dUt is solution to the differential equation
1 2
σ xψ1 x
2
μxψ1 x λψ1 x,
a < x < b,
3.8
with the boundary conditions ψ1 a 0, ψ1 b 1.
∞
Corollary 3.3. The function ψ2 x Ex 0 e−λt dLt is solution to the differential equation
1 2
σ xψ2 x
2
μxψ2 x λψ2 x,
a < x < b,
3.9
with the boundary conditions ψ2 a −1, ψ2 b 0.
For diffusions in Examples 2.3–2.7 we can obtain the explicit expressions for ψ, ψ1 , and ψ2 .
Now we consider the Ornstein-Uhlenbeck process only.
Example 3.4. The Ornstein-Uhlenbeck process is as follows:
dXt vk − Xt dt
σdBt ,
v, σ > 0, k ∈ a, b.
3.10
From Example 2.4, the two independent solutions of differential equation
1 2 σ ψ x
2
vk − xψ x λψx
3.11
are, respectively,
√
√
v
2v
−λ/2v 1/2v/σ 2 x−k2
ψ x H−λ/v −
e
D−λ/v −
x − k 2
x − k ,
σ
σ
√
√
v
2v
−λ/2v 1/2v/σ 2 x−k2
ψ− x H−λ/v
e
D−λ/v
x − k 2
x − k .
σ
σ
3.12
The general solution of 3.11 is of the form
ψx C1 ψ x
C2 ψ− x,
3.13
International Journal of Stochastic Analysis
11
where the constants C1 and C2 are determined by the boundary conditions ψ a θ, ψ b η. They are
C1 θψ b − ηψ a
,
b − ψ aψ− b
ψ− aψ 3.14
ηψ− a − θψ− b
C2 .
ψ− aψ b − ψ aψ− b
4. Applications to Risk Theory
Let Xt denote the surplus of the company. If no dividends were paid, the surplus process
follows the stochastic differential equation
dXt μXt dt
σXt dBt ,
t ≥ 0, X0 x,
4.1
where B is a Brownian motion and μ and σ are Lipschitz-continuous functions.
The company will pay dividends to its shareholders according to barrier strategy with
parameter b > 0. Whenever the surplus is about to go above the level b, the excess will be
paid as dividends, and when the surplus is below b nothing is paid out. Let Dt denote the
aggregate dividends by time t. Thus the resulting surplus process Y is given by
dYt μYt dt
σYt dBt − dDt,
t ≥ 0.
4.2
Let T inf{t ≥ 0 : Yt 0} be the time of ruin. Note that when b < ∞ ruin is certain, that is,
P T < ∞ 1. We are interested in the Laplace transform of T . This model can be found in
Paulsen 14, and some important special cases can be found in Gerber and Shiu 15, Cai et al.
16. It follows from Theorem 2.1 that, for 0 < x < b and λ > 0, Ex e−λT hx, where h is the
solution of
Ahx λhx,
h b 0,
x ∈ 0, b,
4.3
h0 1.
Assume that an insurance company is not allowed to go bankrupt and the beneficiary
of the dividends is required to inject capital into the insurance company to keep its risk
process stays nonnegative. Under such a dividend policy the controlled risk process with
initial reserve x > 0 satisfies
t μ X
t dt
dX
t dBt
σ X
dLt − dDt ,
t ≥ 0,
4.4
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International Journal of Stochastic Analysis
where Lt and Dt are local times at 0 and b, respectively. Lt ensures the insurance company will
not ruin and Dt is the aggregate amount of paid dividends by time t. We consider the total
expected discounted dividends minus the total expected discounted costs of injected capital:
V x; b Ex
∞
e
−λt
dDt − k
∞
0
e−λt dLt ,
4.5
0
where λ > 0 is discounted factor and k is the cost per unit injected capital. Avram et al. 17
consider the problem in a Levy processes setting.
According to Theorem 3.1, we obtain the formula V x, b, as long as we solve the
equation
1 2
σ xV x
2
μxV x λV x,
0 < x < b,
4.6
with the boundary conditions V 0 k, V b 1.
t x} be the first time when the surplus reaches the
For 0 < x < b, let Tx inf{t | X
level x. It follows from Theorem 2.1 that, for 0 < x < b and λ > 0, Ex e−λTx gx, where g is
the solution of
Agx λgx,
g 0 0,
x ∈ 0, b,
4.7
gb 1.
We now give two examples.
Example 4.1. In this example we consider the uncontrolled surplus of insurance company
satisfying Xt x μt σWt , where Wt is Brownian motion. The controlled surplus process X
at time t follows the equation
t x
X
μt
σWt
4.8
Lt − Dt .
Now we consider the differential equation
1 2 σ V x
2
μV x λV x,
0 < x < b,
4.9
with the boundary conditions V 0 k, V b 1. Then
V x, b kesb − 1 rx
kerb − 1 sx
e −
e ,
rb
sb
s e −e
r erb − esb
0 < x < b,
4.10
International Journal of Stochastic Analysis
13
where r and s are the positive root and negative root of the equation σ 2 /2ξ2
respectively, that is,
r
−μ
μ2
2σ 2 λ
σ2
,
−μ −
s
μ2
2σ 2 λ
σ2
.
μξ − λ 0,
4.11
For 0 < x < b and λ > 0, Ex e−λTx : gx is the solution of
1 2 σ g x
2
μg x λgx,
0 < x < b,
4.12
with the boundary conditions g 0 0, gb 1. Solving it gives
Ex e−λTx resx − serx
,
resb − serb
0 < x < b.
4.13
Example 4.2. In this example we consider the Ornstein-Uhlenbeck-type model. The company’s surplus evolves according to
t x
X
t
μ
s ds
ρX
σBt
Lt − Dt .
4.14
0
The model is considered in Cai et al. 16 for the special case where Lt 0.
The diffusion and drift coefficients are σx ≡ σ, μx μ ρx. We consider the
differential equation
1 2 σ V x, b
2
μ
ρx V x, b λV x, b,
0 < x < b,
4.15
with the boundary conditions V 0 k, V b 1. In Cai et al. 16, they pointed out that the
solution is given by
V x, b Ht C1 −t1−c et M1 − a, 2 − c; −t
C2 et Uc − a, c; −t
4.16
for certain coefficients C1 and C2 , with c 1/2, a −λ/2ρ, t −1/ρσ 2 μ ρx2 . Here
M and U are called the confluent hypergeometric functions of the first and second kinds,
respectively. For more details on confluent hypergeometric functions, see Abramowitz and
Stegun 8. It follows from 3.7 in Cai et al. 16 that
H t Ht − C1 1 − c−t−c et M1 − a, 2 − c; −t
− C1
1−a
−t1−c et M2 − a, 3 − c; −t
2−c
C2 c − aet Uc − a
1, c
1; −t.
4.17
14
International Journal of Stochastic Analysis
The conditions C1 and C2 can be determined by V 0 k, V b 1, where V x −2μ
ρx/σ 2 H t. Solving it gives
σ 2 C/ μ
C1 2
2
ρx eμ ρb /ρσ − kD/μ eμ
AD − BC
σ 2 kB/μ eμ
C2 2
2
/ρσ 2
2
− A/ μ ρb eμ
AD − BC
2
/ρσ 2
,
4.18
ρb2 /ρσ 2
,
where
⎛
1/2
−1/2 ⎞ 2
2
μ2
μ
μ
1
⎠
⎝
M 1 − a, 2 − c; 2
A
−
2 ρσ 2
ρσ 2
ρσ
1−a
−
2−c
⎛
μ
B⎝
−
1−a
2−c
μ2
ρσ 2
μ
ρb
ρσ 2
μ2
C U c − a, c; 2
ρσ
D U c − a, c;
μ
ρb
ρσ 2
μ2
M 2 − a, 3 − c; 2
ρσ
2 1/2
ρb
ρσ 2
1/2
1
−
2
2 1/2
μ
ρb
ρσ 2
2 −1/2
M 2 − a, 3 − c;
c − aU c − a
2
,
1, c
⎞
μ
⎠M 1 − a, 2 − c;
2
μ
2
4.19
ρb
,
ρσ 2
μ2
1; 2 ,
ρσ
c − aU c − a
ρb
ρσ 2
1, c
1;
μ
ρb
ρσ 2
2
.
Acknowledgments
The authors thank the reviewer for valuable insights and suggestions that largely contributed
to the improvement of the paper. The research was supported by the National Natural Science
Foundation of China no. 11171179 and the Research Fund for the Doctoral Program of
Higher Education of China no. 20093705110002.
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