Math Meth Oper Res DOI 10.1007/s00186-015-0496-y ORIGINAL ARTICLE Continuity of the optimal average cost in Markov decision chains with small risk-sensitivity Selene Chávez-Rodríguez · Rolando Cavazos-Cadena · Hugo Cruz-Suárez Received: 5 October 2014 / Accepted: 10 February 2015 © Springer-Verlag Berlin Heidelberg 2015 Abstract This note concerns discrete-time controlled Markov chains driven by a decision maker with constant risk-sensitivity λ. Assuming that the system evolves on a denumerable state space and is endowed with a bounded cost function, the paper analyzes the continuity of the optimal average cost with respect to the risk-sensitivity parameter, a property that is promptly seen to be valid at each no-null value of λ. Under standard continuity-compactness conditions, it is shown that a general form of the simultaneous Doeblin condition allows to establish the continuity of the optimal average cost at λ = 0, and explicit examples are given to show that, even if every state is positive recurrent under the action of any stationary policy, the above continuity conclusion can not be ensured under weaker recurrence requirements, as the Lyapunov function condition. Keywords Risk-sensitive optimality equation · Discounted approach · Uniformly bounded solutions to the optimality equation · Jensen’s inequality · Stopping times · Transient states · Continuity at risk-neutrality This work was supported by PSFO under Grant No. 14-300-01, and by PRODEP under Grant No. 17332-UAAAN-CA-23. S. Chávez-Rodríguez · H. Cruz-Suárez Facultad de Ciencias Físico-Matemáticas, Ave. San Claudio y Río Verde, Col. San Manuel CU, Benemérita Universidad Autónoma de Puebla, 72570 Puebla, PUE, Mexico e-mail: selenechavez@alumnos.fcfm.buap.mx H. Cruz-Suárez e-mail: hcs@fcfm.buap.mx R. Cavazos-Cadena (B ) Departamento de Estadística y Cálculo, Universidad Autónoma Agraria Antonio Narro, Boulevard Antonio Narro 1923, Buenavista, 25315 Saltillo, COAH, Mexico e-mail: rcavazos@uaaan.mx 123