Risk-Sensitive and Average Optimality in Markov Decision Processes

This contribution is devoted to the risk-sensitive optimality criteria in finite state Markov Decision Processes. At first, we rederive necessary and sufficient conditions for average optimality of (classical) risk-neutral unichain models. This approach is then extended to the risk-sensitive case, i.e., when expectation of the stream of one-stage costs (or rewards) generated by a Markov chain is evaluated by an exponential utility function. We restrict ourselves on irreducible or unichain Markov models where risk-sensitive average optimality is independent of the starting state. As we show this problem is closely related to solution of (nonlinear) Poissonian equations and their connections with nonnegative matrices.