Appl Math Optim (2010) 61: 167–190 DOI 10.1007/s00245-009-9080-2 The Discounted Method and Equivalence of Average Criteria for Risk-Sensitive Markov Decision Processes on Borel Spaces Rolando Cavazos-Cadena · Francisco Salem-Silva Published online: 30 June 2009 © Springer Science+Business Media, LLC 2009 Abstract This note concerns discrete-time controlled Markov chains with Borel state and action spaces. Given a nonnegative cost function, the performance of a control policy is measured by the superior limit risk-sensitive average criterion associated with a constant and positive risk sensitivity coefficient. Within such a framework, the discounted approach is used (a) to establish the existence of solutions for the corre- sponding optimality inequality, and (b) to show that, under mild conditions on the cost function, the optimal value functions corresponding to the superior and inferior limit average criteria coincide on a certain subset of the state space. The approach of the paper relies on standard dynamic programming ideas and on a simple analytical derivation of a Tauberian relation. Keywords Hölder’s inequality · Contractive operators · Generalized Fatou’s lemma · Risk-sensitive vanishing discount approach · Weak continuity 1 Introduction This work concerns discrete-time Markov decision processes (MDPs) evolving on a Borel space. The system is driven by a risk-averse decision maker with constant risk Dedicated to Professor Onésimo Hernández-Lerma, on the occasion of his sixtieth birthday. This work was supported by the PSF Organization under Grant Np. 08-06(450) and in part by CONACYT under Grant 25357. R. Cavazos-Cadena ( ) Departamento de Estadística y Cálculo, Universidad Autónoma Agraria Antonio Narro, Buenavista, Saltillo, COAH 25315, Mexico e-mail: rcavazos@uaaan.mx F. Salem-Silva Facultad de Matemáticas, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán s/n, Zona Universitaria, Xalapa, VER 91000, Mexico e-mail: frsalem@uv.mx