Necessary and sufficient conditions for a solution to the risk-sensitive Poisson equation on a finite state space

A Markov chain with finite state space endowed with a cost function is considered. The transition mechanism is stationary, the observer has a constant risk-sensitivity, and the overall performance of the chain is measured by the risk-sensitive long-run average cost. In this context, the existence of solutions of the corresponding Poisson equation for arbitrary cost function is characterized in terms of the communication properties of the transition matrix. The result in this direction establishes that the Poisson equation has a solution for each cost function if, and only if, the transition matrix has a unique recurrent class and a strong form of the Doeblin condition holds.