Risk Probability Minimization Problems for Continuous-Time Markov Decision Processes on Finite Horizon

This article deals with a risk probability minimization problem for finite horizon continuous-time Markov decision processes with unbounded transition rates and history-dependent policies. Only using the assumption of nonexplosion of the controlled state process as well as the finiteness of actions available at each state, we not only establish the existence and uniqueness of a solution to the corresponding optimality equation, but also prove the existence of a risk probability optimal policy. Finally, we give two examples to illustrate our results: one example shows that the value iteration algorithm is provided for computing both the value function and an optimal risk probability policy, and the other shows the differences between the conditions in this article and those in the previous literature.