Geodesic
On forward and backward Kolmogorov equations for purely jump Markov processes and their generalizations
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 796-812

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In the present paper, we first give a survey of the forward and backward Kolmogorov equations for pure jump Markov processes with finite and countable state spaces, and then describe relevant results for the case of Markov processes with values in standard Borel spaces based on results of W. Feller and the authors of the present paper.
Keywords: pure jump Markov process, finite and countable state spaces, standard Borel space, differentiability of the transition probability, forward and backward equations, uniqueness of the solution. 
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     author = {E. A. Feinberg and A. N. Shiryaev},
     title = {On forward and backward {Kolmogorov} equations for purely jump {Markov} processes and their generalizations},
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E. A. Feinberg; A. N. Shiryaev. On forward and backward Kolmogorov equations for purely jump Markov processes and their generalizations. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 796-812. http://geodesic.mathdoc.fr/item/TVP_2023_68_4_a7/