Controlled Markov chains with risk-sensitive average cost criterion

Abstract

Discrete controlled Markov chains with finite action space and bounded cost per stage are studied in this dissertation. The performance index function, the exponential average cost (EAC), models risk-sensitivity by means of an exponential (dis)utility function. First, for the finite state space model, the EAC corresponding to a fixed stationary (deterministic) policy is characterized in terms of the spectral radii of matrices associated to irreducible communicating classes of both recurrent and transient states. This result generalizes a well known theorem of Howard and Matheson, which treats the particular case in which the Markov cost chain has only one dosed class of states. Then, it is shown that under strong recurrence conditions, the risk-sensitive model approaches the risk-null model when the risk-sensitivity coefficient is small. However, it is proved and also illustrated by means of examples, that in general, fundamental differences arise between both models, e.g., the EAC may depend on the cost structure at the transient states. In particular, the behavior of the EAC for large risk-sensitivity is also analyzed. After showing that an exponential average optimality equation holds for the countable state space model, a proof of the existence of solutions to that equation for the finite model under a simultaneous Doeblin condition is provided, which is simpler than that given in recent work of Cavazos-Cadena and Fernandez-Gaucherand. The adverse impact of "large risk-sensitivity" on recently obtained conditions for the existence of solutions to an optimality inequality is illustrated by means of an example. Finally, a counterexample is included to show that, unlike previous results for finite models, a controlled Markov chain with infinite state space may not have ultimately stationary optimal policies in the risk-sensitive (exponential) discounted cost case, in general. Moreover, a simultaneous Doeblin condition is satisfied in our example, an assumption that enables the vanishing discount approach in the risk-null case, thus further suggesting that more restrictive conditions than those commonly used in the risk neutral context are needed to develop the mentioned approach for risk-sensitive criteria.