#### Abstract

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 be a one-dimensional time-homogeneous reflected diffusion process with barriers and , which is defined by the following stochastic differential equation: where is a Brownian motion in , and are the regulators of point and , respectively. Further, the processes and are uniquely determined by the following properties (see, e.g., [4]):(1)both and are continuous nondecreasing processes with ,(2) and increase only when and , respectively, that is, and , for .

It is well known that under certain mild regularity conditions on the coefficients and , the SDE (1.1) has a unique strong solution for each starting point (see, e.g., [5]). The solution is a time-homogeneous strong Markov process with infinitesimal generator acting on functions on subject to boundary conditions: .

Define the first passage time where if never reaches .

For , , , we consider the Laplace transform , and the value functions , , and on :

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 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 defined by (1.1).

Theorem 2.1. *Let , , and assume that , satisfy the following equations, respectively:
**If for and for , then
*

*Proof. *Applying the Itô formula for semimartingales to with we obtain
Since is a stopping time and , it follows from the optional sampling theorem that
By the definitions of , , and we have
Substituting them into (2.4) one gets
The result follows.

*Remark 2.2. *Although neither nor 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: , where is a real number.

We consider the differential equation , . It is well known that the increasing and decreasing solutions are, respectively:
where and and are the usual modified Bessel functions.

Then, we can give as follows:
where the constants , and , can be derived from and , respectively. We can obtain their ratios, respectively:
Substituting them into (2.2), we get

*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 .

In the case , , the two independent solutions to are
where and 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 , and , , respectively:
Substituting them into (2.2), we get
For the general and , the two independent solutions are, respectively
Then, as the way used in Example 2.3, we obtain the ratios of the constants , and , , respectively
Substituting them into (2.2), we get
where .

*Remark 2.5. *If we take , , and substitute the series forms of and 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
If is not an integer, the two linear independent solutions are
where and 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 , and , , respectively:
Substituting them into (2.2), we get

*Example 2.7. *The Gompertz Brownian motion process [10] is as follows:

Now consider the differential equation
The increasing and decreasing solutions are, respectively:
where and , 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 , and , , respectively:
Substituting them into (2.2), we get
where denotes .

*Remark 2.8. *For a certain choice of parameters for and 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 or 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 .

Theorem 3.1. *The function defined by (1.5) satisfies the differential equation
**
with the boundary conditions , .*

* Proof. *Applying the Itô’s formula for semimartingales to with we obtain
where we have used that and . From (3.2) we have

Let be a solution of
In place of (3.3), we have
Letting , we get
Likewise
Letting in (3.3) and noting (3.6) and (3.7), we get the desired result.

Corollary 3.2. *The function is solution to the differential equation
**
with the boundary conditions .*

Corollary 3.3. *The function is solution to the differential equation
**
with the boundary conditions , .*

For diffusions in Examples 2.3–2.7 we can obtain the explicit expressions for , , and . 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,
The general solution of (3.11) is of the form
where the constants and are determined by the boundary conditions , . They are

#### 4. Applications to Risk Theory

Let denote the surplus of the company. If no dividends were paid, the surplus process follows the stochastic differential equation where is a Brownian motion and and are Lipschitz-continuous functions.

The company will pay dividends to its shareholders according to barrier strategy with parameter . Whenever the surplus is about to go above the level , the excess will be paid as dividends, and when the surplus is below nothing is paid out. Let denote the aggregate dividends by time . Thus the resulting surplus process is given by Let be the time of ruin. Note that when ruin is certain, that is, . We are interested in the Laplace transform of . 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 and , , where is the solution of

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 satisfies where and are local times at 0 and , respectively. ensures the insurance company will not ruin and is the aggregate amount of paid dividends by time . We consider the total expected discounted dividends minus the total expected discounted costs of injected capital: where is discounted factor and 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 , as long as we solve the equation with the boundary conditions , .

For , let be the first time when the surplus reaches the level . It follows from Theorem 2.1 that, for and , , where is the solution of

We now give two examples.

*Example 4.1. *In this example we consider the uncontrolled surplus of insurance company satisfying , where is Brownian motion. The controlled surplus process at time follows the equation

Now we consider the differential equation
with the boundary conditions , . Then
where and are the positive root and negative root of the equation , respectively, that is,

For and , is the solution of
with the boundary conditions , . Solving it gives

*Example 4.2. *In this example we consider the Ornstein-Uhlenbeck-type model. The company’s surplus evolves according to
The model is considered in Cai et al. [16] for the special case where .

The diffusion and drift coefficients are , . We consider the differential equation
with the boundary conditions . In Cai et al. [16], they pointed out that the solution is given by
for certain coefficients and , with , , . Here and 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
The conditions and can be determined by , , where . Solving it gives
where

#### 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).