This paper studies the minimizing risk problems in Markov decision processes with countable state space and reward set. The objective is to find a policy which Ž . minimizes the probability risk that the total discounted rewards do not exceed a Ž . specified value target . In this sort of model, the decision made by the decision maker depends not only on system's states, but also on his target values. By introducing the decision-maker's state, we formulate a framework for minimizing risk models. The policies discussed depend on target values and the rewards may be arbitrary real numbers. For the finite horizon model, the main results obtained Ž . Ž . are: i The optimal value functions are distribution functions of the target, ii Ž . there exists an optimal deterministic Markov policy, and iii a policy is optimal if and only if at each realizable state it always takes optimal action. In addition, we obtain a sufficient condition and a necessary condition for the existence of finite horizon optimal policy independent of targets and we give an algorithm computing finite horizon optimal policies and optimal value functions. For an infinite horizon model, we establish the optimality equation and we obtain the structure property of optimal policy. We prove that the optimal value function is a distribution function of target and we present a new approximation formula which is the generalization of the nonnegative rewards cases. An example which illustrates the mistakes of previous literature shows that the existence of optimal policy has not been proved really. In this paper, we give an existence condition, which is a sufficient and necessary condition for the existence of an infinite horizon optimal policy independent of targets, and we point out that whether there exists an optimal policy remains an open problem in the general case.

No abstract