Revisiting the forward equations for inhomogeneous semi-Markov processes

TL;DR

This paper revisits the forward equations for inhomogeneous semi-Markov processes, establishing new theoretical results on transition intensities, probabilities, and their relationships using marked point process techniques.

Contribution

It provides rigorous proofs connecting transition intensities with transition probabilities and clarifies relationships among various semi-Markov process classes.

Findings

Transition intensities are right-derivatives of transition probabilities in countably infinite states.

Transition probabilities satisfy forward equations under certain regularity conditions.

Relationships between different semi-Markov process classes are established.

Abstract

In this paper, we consider a class of inhomogeneous semi-Markov processes directly based on intensity processes for marked point processes. We show that this class satisfies the semi-Markov properties defined elsewhere in the literature. We use the marked point process setting to derive strong upper bounds on various probabilities for semi-Markov processes. Using these bounds, we rigorously prove for the case of countably infinite state space that the transition intensities are right-derivatives of the transition probabilities, and we prove for the case of finite state space that the transition probabilities satisfy the forward equations, requiring only right-continuity of the transition intensities in the time and duration arguments and a boundedness condition. We also show relationships between several classes of semi-Markov processes considered in the literature, and we prove an integral representation for the left derivatives of the transition probabilities in the duration parameter.