Some Study On Occupation Density Of Diffusion Processes

We consider the joint Laplace transform of occupation time and local time over finite interval(0,a),(a,b)for stochastic processes.These two topics are interesting for researchers.According to the conclusions in precious studies,we already know the results on pre-exit or two-side exit joint occupation times for spectrally negative Levy processes(SNLP)and diffusion processes.Recently.Li et al.(2014)adopted the Poisson approach in Li and Zhou(2014)to obtaiin expressioin of double Laplace transform for diffusion processes.It has been found that these results have many applications in mathematical finance and risk theory for insurance.In mathematical finance,the occupation times can be used to define price options such as step options and corridor options.In risk theory,the occupation times can be utilized as a useful tool to manage insurable risks.This paper has three chapters.The first chapter is an introduction.Firstly,we introduce the research background and both the domestic and international research profile in this filed.Then we outline the main results.In the second chapter,we briefly introduce definitions and related properties about time-homogeneous diffusion processes.Meanwhile,we also discuss the background,significance and current situation.Furthermore,it gives a general introduction of definitions and related properties of the Brownian motion with drift.Based on the previous study we find expressions of the Laplace transform for occupation times before the exponential stopping time eq,over joint intervals(0,a),(a,b).In the third chapter,we compute the Laplace transform of occupation density for the Brownian motion with drift.On the basis of the second chapter,we adopt the Poisson approach of Li and Zhou(2014)to obtain the expressions of the Laplace transform for occupation density.Under the condition stated before,we compute expressions of the Laplace transform for local times before the exponential stopping time eq,over different joint intervals(0,a),(a,b).Their expressions have the forms as followsExe-?l(eq,0),Exe-?l(eq,a).