| Occupation times, local times and potential measures are all interesting and active topics in the study of stochastic processes recently. In particular, several approaches have been proposed in recent years to find Laplace transforms of occupation time, which have found successful applications in risk models for insurance and in mathematical finance models. Based on the recent research on these issues, in this thesis we further study the occupation time, local times and potential measures.This thesis is divided into three chapters. The first chapter is an introduction. Firstly, we introduces the research background and both the domestic and international research profile in this field. We then outline the main results of this thesis.In the second chapter we generalize the known results on Laplace transforms of occupation time for homogenous diffusion process. Using the approximation ap-proach of Landriault et al (2011) and Li and Zhou (2013), for a one-dimensional time-homogeneous diffusion process X and constants c< a< b< d we find expressions of double Laplace transforms of the following form where Tx denotes the first passage time of level x. Then we give an example of appli-cation, in which we find explicit Laplace transforms of the corresponding occupation time and occupation density for the Brownian motion with two-valued drift.In the third chapter, we study the occupation time and the potential measure for spectrally negative Levy processes. Using a new and more direct approach of Li and Zhou (2014) that takes use of a property of Poisson process to obtain Laplace transforms of occupation times, we find expressions of potential measures that are discounted by their joint occupation times over scmi-infinite intervals (-∞,0) and (0, ∞). The expressions of Laplace transforms are similarly in terms of the associated scale functions and the inverse functions of Laplace exponents for the spectrally neg-ative Levy processes. At the end of this chapter, we apply these results to find more explicit expressions for the examples of Brownian motion and stable spectrally negative process. Two research papers have been published based on this thesis. |
