  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) 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, which motivated this work in the first place.
