Solution to the risk-sensitive average optimality equation in communicating Markov decision chains with finite state space: An alternative approach

This note concerns Markov decision chains with finite state and action sets. The decision maker is assumed to be risk-averse with constant risk-sensitive coefficient λ, and the performance of a control policy is measured by the risk-sensitive average cost criterion. In their seminal paper Howard and Matheson established that, when the whole state space is a communicating class under the action of each stationary policy, then there exists a solution to the optimality equation for every λ > 0. This paper presents an alternative proof of this fundamental result, which explicitly highlights the essential role of the communication properties in the analysis of the risk-sensitive average cost criterion.