Continuous-Time Controlled Markov Chains with Discounted Rewards

This paper studies denumerable state continuous-time controlled Markov chains with the discounted reward criterion and a Borel action space. The reward and transition rates are unbounded, and the reward rates are allowed to take positive or negative values. First, we present new conditions for a nonhomogeneous Q(t)-process to be regular. Then, using these conditions, we give a new set of mild hypotheses that ensure the existence of -optimal (0) stationary policies. We also present a martingale characterization of an optimal stationary policy. Our results are illustrated with controlled birth and death processes.