Constrained Markov control processes with randomized discounted cost criteria: Occupation measures and extremal points

In this paper we study constrained Markov control processes on Borel spaces with possibly unbounded costs, under a discounted optimality criterion with random discount factor and restrictions of the same kind. Imposing mild conditions, we show the solubility of the corresponding control problem. Furthermore, we characterize the corresponding occupation measures and show the existence of randomized optimal control policies. These optimal strategies are convex combinations of deterministic stationary policies.