The risk probability criterion for discounted continuous-time Markov decision processes

In this paper, we consider the risk probability minimization problem for infinite discounted continuous-time Markov decision processes (CTMDPs) with unbounded transition rates. First, we introduce a class of policies depending on histories with the additional reward levels. Then, we construct the corresponding probability spaces, and also establish the non-explosion of the state process. Secondly, under suitable conditions we prove that the value function is a solution to the optimality equation for the probability criterion by an iteration technique, and obtain a value iteration algorithm to compute (at least approximate) the value function. Furthermore, under an additional condition we establish the uniqueness of the solution to the optimality equation and the existence of an optimal policy. Finally, we illustrate our results with two examples. The first one is used to verify our conditions for CTMDPs with unbounded transition rates, the second one for the numerical calculation of the value function and an optimal policy.