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

Chuancun Yin,Huiqing Wang

doi:10.1155/2012/971212

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.

Highlights

Introduction and the ModelDiffusion processes with one or two barriers appear in many applications in economics, finance, queueing, mathematical biology, and electrical engineering
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 μ Xt dt σ Xt dBt dLt − dUt, 1.1
X0 x ∈ a, b, International Journal of Stochastic Analysis where B t is a Brownian motion in R, L {Lt, t ≥ 0} and U {Ut, t ≥ 0} are the regulators of point a and b, respectively

Summary

Introduction and the Model

Diffusion processes with one or two barriers appear in many applications in economics, finance, queueing, mathematical biology, and electrical engineering. Motivated by Ward and Glynn’s one-sided problem, Bo et al 3 considered a reflected Ornstein-Uhlenbeck process with two-sided barriers. 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 μ Xt dt σ Xt dBt dLt − dUt, 1.1. The solution Xt is a time-homogeneous strong Markov process with infinitesimal generator. 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 :.

Laplace Transform

The Value Function

Applications to Risk Theory