Research On Occupation Times Of Diffusion Processes

Occupation time,the amount of time a stochastic process spends in a cer-tain region.In 1940,Levy[62]firstly found the distribution of occupation time for standard Brownian motion,which stays in positive or negative half axis.That is the classical arc-sine law.Since then,many domestic and foreign schol-ars have studied occupation times theory,and adopted different approaches to finding Laplace transform of occupation times for stochastic processes.The study is of theoretical interest and finds many applications in risk theory and finance.In this paper,we adopt the Poisson approach of Li and Zhou[65]to con-sider the joint Laplace transform of occupation times for diffusion processes,which stays in two disjoint intervals.The innovation of this paper is as follow:(1)Deriving the joint Laplace transforms of occupation times in(a,r)and(r,b)of diffusion processes up to the two-sided exit time.(2)Finding the expres-sions of two-sided discounted potential measures for diffusion,which stay in two disjoint intervals(-?,a)and(a,+?).(3)Obtaining explicit expressions on the joint Laplace transforms of occupation times for several examples of diffusion processes.The organization is as below:In chapter 1,we will briefly introduce the research status about occupation time at home and abroad and the research significance.Some preliminaries will be introduced,such as the related concepts of diffusion process and two-sided exit problem.In chapter 2,we will adopt the approach of Li and Zhou[65],excursion theory and strong Markov property to consider the joint occupation times in(a,r)and(r,b)of diffusion processes up to the two-sided exit time.The expressions of Laplace transform and potential measure will be given.As the application of the second chapter,in Chapter 3,we will apply the results in the previous chapter to find explicit expressions on the joint Laplace transforms of occupation times for several examples of diffusion processes,such as Brownian motion with drift,Brownian motion with alternating drift and skew Brownian motion.In the fourth chapter,we continue to consider the two-sided discounted potential measures for diffusion,which stay in two disjoint intervals(-?,a)and(a,+?).The expressions of potential measure will be obtained.In the chapter 5,we apply the results in chapter 4 to compute explicit expressions on two-sided discounted potential measures for several examples of diffusion processes,such as Brownian motion with drift,skew Brownian motion and Brownian motion with two-valued drift.Finally,we introduce the follow-on work briefly.