The minimal hitting probability of continuous-time controlled Markov systems with countable states

This paper concentrates on the minimal hitting probability of continuous-time controlled Markov systems (CTCMSs) with countable state and finite admissible action spaces. The existence of an optimal policy is first proved. In particular, for a special and important case of controlled branching processes (CBPs), it is proved that the minimal hitting probability is the unique solution to an improved optimal system of equations. Furthermore, a novel and precise improved-policy iteration algorithm of an optimal policy and the minimal hitting probability (minimal extinction probability) is presented for CBPs.