A study of Component-Wise Semi-Markov Process

Abstract

Component-wise semi-Markov processes (CSM) constitute a larger class of pure jump processes which includes semi-Markov, and Markov pure jump processes. This thesis examines semi-Markov as well as CSM processes with dependent components. In order to better understand the interactions among components of CSM processes having bounded transition rates, we consider a family of stochastic flows using a system of SDEs driven by Poisson random measure (PRM), with an additional gaping parameter. More specifically, we have demonstrated that the proposed system of SDEs driven by a PRM does, in fact, has a unique solution. Then, we prove that a solution satisfies the desired law. Thus we establish a semimartingale representation of the homogeneous or nonhomogeneous semi-Markv process. Finally, we pick up an appropriate flow by fixing the gaping parameter. We derive expressions of the probabilities of meeting and merging of a pair of semi-Markov processes, solving the same equation but with different initial conditions. We also obtain a set of sufficient