Research On Occupation Times Of Several Classes Of Markov Processes

The occupation time is the amount of time a stochastic process stays within a certain range.It is of great theoretical interest and finds many applications in risk theory and finance.Several approaches have been proposed in recent years to find Laplace transforms of occupation time,which have found successful application in risk models for insurance and in mathematical finance models.In this paper,we adopt the Poisson approach of Li and Zhou(2014)that takes use of a property of Poisson process to consider the joint Laplace transform of occupation times for diffusion processes which stays in n disjoint intervals.The expressions are in terms of solutions to the associated differential equations.And then,according to the same approach,we find expressions of potential measures that are discounted by their occupation times over a finite intervals for spectrally negative levy processes.The expressions are in terms of the associated scale functions and the inverse functions of Laplace exponents for spectrally negative levy processes.At the end,we apply these results to find more explicit expression for the example.