Existence and Regularity of a Nonhomogeneous Transition Matrix under Measurability Conditions

Liuer Ye (), Xianping Guo () and Onésimo Hernández-Lerma ()
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Liuer Ye: Zhongshan University
Xianping Guo: Zhongshan University
Onésimo Hernández-Lerma: CINVESTAV-IPN

Journal of Theoretical Probability, 2008, vol. 21, issue 3, 604-627

Abstract: 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≥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.

Keywords: Nonhomogeneous continuous-time Markov chains; Nonhomogeneous transition rates; Kolmogorov equations; Minimum transition matrix; 60J27; 60J35; 60J75 (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-008-0163-9

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