Minimizing Risk Models in Denumerable Semi-Markov Decision Processes with a Target Set

This paper deals with the risk minimization problem in semi-Markov decision processes with denumerable states. The criterion to be minimized is the risk probability that a total reward over a first passage time to some target set doesn't exceed a level. We first characterize the optimal value function, and then establish the optimality equation and the existence of optimal policies under mild conditions. Moreover, we give some sufficient conditions for the existence of an optimal policy, and these conditions are imposed on the primitive data of the model and are thus easy to verify. Finally, a numerical example is given to illustrate our results.