On solutions of Kolmogorov's equations for jump Markov processes

TL;DR

This paper explores three methods to construct nonhomogeneous jump Markov processes and proves their equivalence and uniqueness under certain conditions, extending classical results to more general Q-functions.

Contribution

It demonstrates the equivalence of three constructions of jump Markov processes and extends Kolmogorov equation solutions to measurable Q-functions.

Findings

All three constructions define the same transition function.

Unique solutions to Kolmogorov equations exist for regular transition functions.

Extension of Feller's results to measurable Q-functions.

Abstract

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller's seminal paper. In particular, this paper extends Feller's results for continuous Q-functions to measurable Q-functions and provides additional results.