Optimality equations and inequalities in a class of risk-sensitive average cost Markov decision chains

This note concerns controlled Markov chains on a denumerable sate space. The performance of a control policy is measured by the risk-sensitive average criterion, and it is assumed that (a) the simultaneous Doeblin condition holds, and (b) the system is communicating under the action of each stationary policy. If the cost function is bounded below, it is established that the optimal average cost is characterized by an optimality inequality, and it is to shown that, even for bounded costs, such an inequality may be strict at every state. Also, for a nonnegative cost function with compact support, the existence an uniqueness of bounded solutions of the optimality equation is proved, and an example is provided to show that such a conclusion generally fails when the cost is negative at some state.