On weak conditions and optimality inequality solutions in risk-sensitive controlled Markov processes with average criterion

A standard approach to the problem of finding optimal policies for controlled Markov processes with average cost is based on the existence of solutions to an average optimality equation, or an average optimality inequality (Cavazos-Cadena and Sennott (1992), Sennott (1999)). In the latter, conditions are imposed on the solutions to the inequalities such that if one such solution is found, then optimal policies are obtained for all values of the state. In Hernandez-Lerma and Lasserre, (1994), such conditions are relaxed, at the expense that perhaps optimal policies are characterized for only a proper subset of the state space. Motivated by the work in Hernandez-Lerma and Lasserre, optimality inequality results were presented in Hernandez-Hernandez and Marcus, (1999), for the risk-sensitive case, purposely trying to emulate in the risk-sensitive case what had been done previously for the risk-neutral case. However, as it is illustrated in the sequel, the results in Hernandez-Hernandez and Marcus exhibit an acute fragility not present in their risk- counterparts.