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Existence and Regularity of a Nonhomogeneous Transition Matrix under Measurability Conditions

Author

Listed:
  • Liuer Ye

    (Zhongshan University)

  • Xianping Guo

    (Zhongshan University)

  • Onésimo Hernández-Lerma

    (CINVESTAV-IPN)

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.

Suggested Citation

  • Liuer Ye & Xianping Guo & Onésimo Hernández-Lerma, 2008. "Existence and Regularity of a Nonhomogeneous Transition Matrix under Measurability Conditions," Journal of Theoretical Probability, Springer, vol. 21(3), pages 604-627, September.
    • Handle: RePEc:spr:jotpro:v:21:y:2008:i:3:d:10.1007_s10959-008-0163-9
    DOI: 10.1007/s10959-008-0163-9
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    Cited by:

    1. Eugene Feinberg & Manasa Mandava & Albert N. Shiryaev, 2022. "Kolmogorov’s equations for jump Markov processes with unbounded jump rates," Annals of Operations Research, Springer, vol. 317(2), pages 587-604, October.

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