Constrained discounted Markov decision processes with Borel state spaces

We study discrete-time discounted constrained Markov decision processes (CMDPs) with Borel state and action spaces. These CMDPs satisfy either weak (W) continuity conditions, that is, the transition probability is weakly continuous and the reward function is upper semicontinuous in state–action pairs, or setwise (S) continuity conditions, that is, the transition probability is setwise continuous and the reward function is upper semicontinuous in actions. Our main goal is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of randomized stationary policies and in the class of chattering policies introduced in this paper. If the initial distribution and transition probabilities are atomless, then using a general “purification result” of Feinberg and Piunovskiy we show the existence of a deterministic (stationary) optimal policy. Our main results are illustrated by examples.