Average optimality for continuous-time Markov decision processes under weak continuity conditions

This article considers the average optimality for a continuous-time Markov decision process with Borel state and action spaces and an arbitrarily unbounded nonnegative cost rate. The existence of a deterministic stationary optimal policy is proved under a different and general set of conditions as compared to the previous literature; the controlled process can be explosive, the transition rates can be arbitrarily unbounded and are weakly continuous, the multifunction defining the admissible action spaces can be neither compact-valued nor upper semi-continuous, and the cost rate is not necessarily inf-compact

Continuous-time Markov decision processes under the risk-sensitive average cost criterion

This paper studies continuous-time Markov decision processes under the risk-sensitive average cost criterion. The state space is a finite set, the action space is a Borel space, the cost and transition rates are bounded, and the risk-sensitivity coefficient can take arbitrary positive real numbers. Under the mild conditions, we develop a new approach to establish the existence of a solution to the risk-sensitive average cost optimality equation and obtain the existence of an optimal deterministic stationary policy.Comment: 14 page

Average optimality for continuous-time Markov decision processes in polish spaces

This paper is devoted to studying the average optimality in continuous-time Markov decision processes with fairly general state and action spaces. The criterion to be maximized is expected average rewards. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We first provide two optimality inequalities with opposed directions, and also give suitable conditions under which the existence of solutions to the two optimality inequalities is ensured. Then, from the two optimality inequalities we prove the existence of optimal (deterministic) stationary policies by using the Dynkin formula. Moreover, we present a ``semimartingale characterization'' of an optimal stationary policy. Finally, we use a generalized Potlach process with control to illustrate the difference between our conditions and those in the previous literature, and then further apply our results to average optimal control problems of generalized birth--death systems, upwardly skip-free processes and two queueing systems. The approach developed in this paper is slightly different from the ``optimality inequality approach'' widely used in the previous literature.Comment: Published at http://dx.doi.org/10.1214/105051606000000105 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org