A poisson equation for the risk-sensitive average cost in semi-markov chains

Abstract

This work concerns semi-Markov chains evolving on a finite state space. The system development generates a cost when a transition is announced, as well as a holding cost which is incurred continuously during each sojourn time. It is assumed that these costs are paid by an observer with positive and constant risk-sensitivity, and the overall performance of the system is measured by the corresponding (long-run) risk-sensitive average cost criterion. In this framework, conditions are provided under which the average index does not depend on the initial state and is characterized in terms of a single Poisson equation.