Existence of risk sensitive optimal stationary policies for controlled Markov processes.pdf

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Existence of risk sensitive optimal stationary policies for controlled Markov processes.pdf

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Existence of risk sensitive optimal stationary policies for controlled Markov processes

Existence of Risk Sensitive Optimal StationaryPolicies for Controlled Markov Processes1 Daniel Hernandez-Hernandez2 and Steven I. Marcus3 AbstractIn this paper we are concerned with the existence of optimal stationary poli-cies for innite horizon risk sensitive Markov control processes with denu-merable state space, unbounded cost function, and long run average cost.Introducing a discounted cost dynamic game, we prove that its value func-tion satises an Isaacs equation, and its relationship with the risk sensitivecontrol problem is studied. Using the vanishing discount approach, we provethat the risk- sensitive dynamic programming inequality holds, and derivean optimal stationary policy.Key Words. Risk sensitive stochastic control, dynamic games, Isaacs equa-tion, optimal stationary policies.Mathematics Subject clasications (1991). 90C40 (93E20).Running Head. Risk Sensitive Controlled Markov Processes.1Supported in part by the National Science Foundation under grant EECnstitute for Systems Research, University of Maryland, College Park, Maryland20742. On leave from Department of Mathematics, CINVESTAV-IPN, MEXICO,dher@math.cinvestav.mx3Electrical Engineering Department and Institute for Systems Research, University ofMaryland, College Park, Maryland 20742, marcus@src.umd.edu1 1 IntroductionIn this paper we are concerned with the existence of optimal stationary poli-cies for innite horizon risk sensitive stochastic control problems with denu-merable state space, discrete time parameter, unbounded cost function, andlong run average cost. For the risk neutral stochastic control problem, thesame kind of problem has been addressed, see e.g. [CC, CC-S, S1, S2, HL-L1, HL-L2], exploiting the vanishing discount approach, in which the valuefunction of the average cost control problem is approximated by the valuefunction of a sequence of discounted problems. However, for the risk sensitivecontrol problem there does not seem to be a sequence of discounted contro