Existence and Regularity of a Nonhomogeneous Transition Matrix under Measurability Conditions

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≥0) 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, and this was the main motivation for this work. Supported by NSFC and RFDP. The research of O. Hernández-Lerma was partially supported by CONACYT grant 45693-F.