Existence and regularity of a nonhomogeneous transition matrix under measurability conditions

TL;DR

This paper establishes the existence and regularity of transition matrices for nonhomogeneous continuous-time Markov processes under mild measurability conditions, expanding applicability in stochastic control and game theory.

Contribution

It extends the theory by allowing non-conservative and merely measurable rate matrices, providing necessary and sufficient conditions for regularity.

Findings

Transition matrices exist under mild measurability conditions.

The constructed transition matrix is minimal.

A criterion for regularity of the transition matrix is provided.

Abstract

This paper is about the existence and regularity of the transition probability matrix of a nonhomogeneous continuous-time Markov process with a countable state space. A standard approach to prove the existence of such a transition matrix is to begin with a continuous (in t) and conservative matrix Q(t)=[q_{ij}(t)] of nonhomogeneous transition rates q_{ij}(t), and use it to construct the transition probability matrix. Here we obtain the same result except that the q_{ij}(t) are only required to satisfy a mild measurability condition, and Q(t) may not be conservative. Moreover, the resulting transition matrix is shown to be the minimum transition matrix and, in addition, a necessary and sufficient condition for it to be regular is obtained. These results are crucial in some applications of nonhomogeneous continuous-time Markov processes, such as stochastic optimal control problems and stochastic games, which motivated this work in the first place.