Kolmogorov's Equations for Jump Markov Processes with Unbounded Jump Rates

TL;DR

This paper extends the understanding of Kolmogorov's equations for jump Markov processes by establishing conditions under which the minimal solutions correspond to the transition probabilities, even with unbounded jump rates.

Contribution

It generalizes previous results by providing new sufficient conditions for the minimal solutions of Kolmogorov's equations to match transition probabilities in cases of unbounded jump rates.

Findings

Minimal solution of backward equation matches transition probability under local integrability.

Minimal solution of forward equation matches transition probability under local boundedness.

Extends classical results to more general unbounded rate scenarios.

Abstract

As well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's forward equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.