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 It？ 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