Theory of Probability &Its Applications, Volume 66, Issue 4, Page 582-600, February 2022.
This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference “P. L. Chebyshev -- 200,” and it describes the results of their joint studies with Manasa Mandava (1984--2019).

中文翻译：

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跳跃马尔可夫过程的 Kolmogorov 方程及其在控制问题中的应用

概率理论及其应用，第 66 卷，第 4 期，第 582-600 页，2022 年 2 月
。本文描述了非齐次跳跃马尔可夫过程的 Kolmogorov 方程组的解的结构，以及这些结果在跳跃随机系统控制中的应用。Feller [Trans。阿米尔。数学。Soc., 48 (1940), pp. 488--515]，他在 1945 年在那篇论文的勘误表中澄清说，它的一些结果只涵盖了非爆炸性马尔可夫过程。在这项主要是调查性质的工作中，还考虑了爆炸过程的情况。本文基于作者在“PL Chebyshev -- 200”会议上发表的特邀演讲，描述了他们与 Manasa Mandava (1984--2019) 联合研究的结果。