A Characterization of the optimal risk-Sensitive average cost in finite   controlled Markov chains
  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. 
Rolando Cavazos-Cadena and Daniel Hernandez-Hernandez