The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

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.


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 {X t , 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: dX t μ X t dt σ X t dB t dL t − dU t , X 0 x ∈ a, b , International Journal of Stochastic Analysis where B t is a Brownian motion in R, L {L t , t ≥ 0} and U {U t , 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 → L t and t → U t are continuous nondecreasing processes with L 0 U 0 0, 2 L and U increase only when X a and X b, respectively, that is, It is well known that under certain mild regularity conditions on the coefficients μ x and σ x , the SDE 1.1 has a unique strong solution for each starting point see, e.g., 5 . The solution X t is a time-homogeneous strong Markov process with infinitesimal generator acting on functions on a, b subject to boundary conditions: f a f b 0. Define the first passage time where τ y ∞ if X t never reaches y.
For λ > 0, η > 0, θ > 0, we consider the Laplace transform ϕ, and the value functions ψ, ψ 1 , and ψ 2 on x ∈ a, b : 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. Bo et al. 3 consider the Laplace transform E x e −λτ y 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 .

Laplace Transform
International Journal of Stochastic Analysis 3 Theorem 2.1. Let x ∈ a, b , λ > 0, and assume that f 1 y , f 2 y satisfy the following equations, respectively: Af 2 y λf 2 y , y ∈ a, b ,

2.1
If f 1 y / 0 for y ∈ x, b and f 2 y / 0 for y ∈ a, x , then

2.2
Proof. Applying the Itô formula for semimartingales to h t,

2.3
Since τ y < ∞ is a stopping time and x ∈ a, b , it follows from the optional sampling theorem that

4 International Journal of Stochastic Analysis
By the definitions of τ y , U, and L we have Substituting them into 2.4 one gets

2.6
The result follows.

Remark 2.2.
Although neither f 1 · nor f 2 · in the Theorem 2.1 is unique, but each of their ratios is unique.
As illustrations of Theorem 2.1, we consider some examples.
where d > 1 is a real number. We consider the differential equation 1/2 ψ x d − 1 /2x ψ x λψ x , λ > 0. It is well known that the increasing and decreasing solutions are, respectively: where ν d − 2 /2 and I ν and K ν are the usual modified Bessel functions. Then, we can give f 1 x , f 2 x as follows: where the constants C 1 , C 2 and C 3 , C 4 can be derived from f 1 a 0 and f 2 b 0, respectively. We can obtain their ratios, respectively:

2.9
International Journal of Stochastic Analysis 5 Substituting them into 2.2 , we get , y ∈ a, x .

2.10
Example 2.4. The Ornstein-Uhlenbeck process 6 is as follows: 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 where H ν · and D ν · 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 C 1 , C 2 and C 3 , C 4 , respectively:

2.13
Substituting them into 2.2 , we get , y ∈ a, x .
2.14 6 International Journal of Stochastic Analysis For the general k and σ, the two independent solutions are, respectively

2.15
Then, as the way used in Example 2.3, we obtain the ratios of the constants C 1 , C 2 and C 3 , C 4 , respectively

2.16
Substituting them into 2.2 , we get
Remark 2.5. If we take k 0, σ 1, a 0 and substitute the series forms of H ν · and D ν · 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 : Now consider the differential equation International Journal of Stochastic Analysis 7 If 2v/σ 2 k is not an integer, the two linear independent solutions are 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 C 1 , C 2 and C 3 , C 4 , respectively:

2.22
Example 2.7. The Gompertz Brownian motion process 10 is as follows: dX t vX t ln k − ln X t dt σX t dB t , v,σ > 0, k ∈ a, b .

2.23
Now consider the differential equation The increasing and decreasing solutions are, respectively: 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 C 1 , C 2 and C 3 , C 4 , respectively:

2.26
Substituting them into 2.2 , we get

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 with the boundary conditions ψ a θ, ψ b η.
Proof. Applying the Itô's formula for semimartingales to h t, X t e −λt ψ X t with ψ ∈ where we have used that ψ X t dL t ψ a dL t and ψ X t dU t ψ b dU t . From 3.2 we have

3.3
Let f be a solution of

3.4
In place of 3.3 , we have Letting t → ∞, we get Letting t → ∞ in 3.3 and noting 3.6 and 3.7 , we get the desired result.
∞ 0 e −λt dU t is solution to the differential equation with the boundary conditions ψ 1 a 0, ψ 1 b 1.

Corollary 3.3. The function ψ 2 x E x
∞ 0 e −λt dL t is solution to the differential equation with the boundary conditions ψ 2 a −1, ψ 2 b 0.
For diffusions in Examples 2.3-2.7 we can obtain the explicit expressions for ψ, ψ 1 , and ψ 2 . Now we consider the Ornstein-Uhlenbeck process only.
Example 3.4. The Ornstein-Uhlenbeck process is as follows: From Example 2.4, the two independent solutions of differential equation are, respectively,

3.12
The general solution of 3.11 is of the form where the constants C 1 and C 2 are determined by the boundary conditions ψ a θ, ψ b η. They are 3.14

Applications to Risk Theory
Let X t denote the surplus of the company. If no dividends were paid, the surplus process follows the stochastic differential equation dX t μ X t dt σ X t dB t , t ≥ 0, X 0 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 D t denote the aggregate dividends by time t. Thus the resulting surplus process Y is given by dY t μ Y t dt σ Y t dB t − dD t , t ≥ 0.

4.2
Let T inf{t ≥ 0 : Y t 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, E x e −λT h x , where h is the solution of