Some Fluctuation Identities of Hyper-Exponential Jump-Diffusion Processes

Meromorphic L´evy processes have attracted the attention of a lot of researchers recently due to its special structure of the Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. With these Wiener-Hopf factors in hand, we can explicitly derive the expression of fluctuation identities that concern the first passage problems for finite and infinite intervals for the meromorphic L´evy process and the resulting process reflected at its infimum. In this thesis, we consider some fluctuation identities of some classes of meromorphic jump-diffusion processes with either the double exponential jumps or more general the hyper-exponential jumps. We study solutions to the one-sided and two-sided exit problems, and potential measure of the process killed on exiting a finite or infinite intervals. Also, we obtain some results to the process reflected at its infimum.