This work concerns controlled Markov chains with finite state and action spaces. The transition law satisfies the simultaneous Doeblin condition, and the performance of a control policy is measured by the (long-run) risk-sensitive average cost criterion associated to a positive, but otherwise arbitrary, risk sensitivity coefficient. Within this context, the optimal risk-sensitive average cost is characterized via a minimization problem in a finite-dimensional Euclidean space.
"A characterization of the optimal risk-sensitive average cost in finite controlled Markov chains." Ann. Appl. Probab. 15 (1A) 175 - 212, February 2005. https://doi.org/10.1214/105051604000000585