  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
(1940), 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 "Chebyshev-200", and it describes the results of their joined
studies with Manasa Mandava (1984-2019).
