Some Study On Occupation Time And The Related Properties Of Diffusion Process

In recent years,the occupation time and exit problems are two hot issues of stochastic process,and these results are widely used in risk theory of insurance and option pricing of financial mathematics.Based on these results.In the thesis we use the Poisson method of Li and Zhou(2014),exit identities and potential measure theory of the diffusion processes to further study the occupation time of the diffusion processes.This thesis includes three chapters.The first chapter introduces the background and related literature,and then sum-marizes the main contents and main results of this thesis.In the second chapter we find the following joint Laplace transform of occupation times over semi-infinite intervals(-?,a)and(b,?)for diffusion process before an independent exponential time eq,Eye-?-?0eq1(?,a)(Xs)ds-?+?0eq(b,?)(Xs)ds,0<a<b.In the third chapter we find the following two-sided exit identities for diffusion process that is only observed at arrival times of an independent Poisson process,Px(?0<Ta+),Ex(e-?[X(Ta+)-a];Ta+<?0),Ex(e?X(T0-);T0-<?a)and Ex[e?x(T0-);T0-<Ta+),Ex[e-?[X(Ta+)-a];Ta+<T0-].We then apply these results to recover more explicit expressions for Brownian motion.