Continuous Semi-Markov Processes and Their Applications

Abstract

A special class of random processes in a metric space of states is considered. Such a process is assumed to have trajectories being continuous on the right and having limits at the left. Moreover, it has the Markov property with respect to any intrinsic stopping time such as a time of the first exit from an open set or any finite iteration of such times. With certain extansion of the term we call such a process semi-Markov. The class of semi-Markov processes (generalized in such a way) includes as subclasses both all strong Markov processes, and also stepped semi-Markov processes of Lévy-Smith. In the article, we deal with non-Markov semi-Markov processes with continuous trajectories and, in particular, semi-Markov processes of a diffusion type. Some general properties of these processes are exposed. Some examples of applications of continuous semi-Markov processes to chromatography and reliability are given.