  As well-known, transition probabilities of jump Markov processes satisfy
Kolmogorov's backward and forward equations. In the seminal 1940 paper, William
Feller investigated solutions of Kolmogorov's equations for jump Markov
processes. Recently the authors solved the problem studied by Feller and showed
that the minimal solution of Kolmogorov's backward and forward equations is the
transition probability of the corresponding jump Markov process if the
transition rate at each state is bounded. This paper presents more general
results. For Kolmogorov's backward equation, the sufficient condition for the
described property of the minimal solution is that the transition rate at each
state is locally integrable, and for Kolmogorov's forward equation the
corresponding sufficient condition is that the transition rate at each state is
locally bounded.
