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Local Poisson Equations Associated with Discrete-Time Markov Control Processes

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Listed:
  • Daniel Hernández Hernández

    (Centro de Investigación en Matemáticas)

  • Diego Hernández Bustos

    (Centro de Investigación en Matemáticas)

Abstract

This paper provides a characterization of the optimal average cost function, when the long-run (risk-sensitive) average cost criterion is used. The Markov control model has a denumerable state space with finite set of actions, and the characterization presented is given in terms of a system of local Poisson equations, which gives as a by-product the existence of an optimal stationary policy.

Suggested Citation

  • Daniel Hernández Hernández & Diego Hernández Bustos, 2017. "Local Poisson Equations Associated with Discrete-Time Markov Control Processes," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 1-29, April.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:1:d:10.1007_s10957-017-1076-5
    DOI: 10.1007/s10957-017-1076-5
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    References listed on IDEAS

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    1. Rolando Cavazos-Cadena, 2003. "Solution to the risk-sensitive average cost optimality equation in a class of Markov decision processes with finite state space," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(2), pages 263-285, May.
    2. Rolando Cavazos-Cadena & Emmanuel Fernández-Gaucherand, 1999. "Controlled Markov chains with risk-sensitive criteria: Average cost, optimality equations, and optimal solutions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(2), pages 299-324, April.
    3. Rolando Cavazos-Cadena & Daniel Hernández-Hernández, 2003. "Solution to the risk-sensitive average optimality equation in communicating Markov decision chains with finite state space: An alternative approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(3), pages 473-479, January.
    4. Karel Sladký & Raúl Montes-de-Oca, 2008. "Risk-Sensitive Average Optimality in Markov Decision Chains," Operations Research Proceedings, in: Jörg Kalcsics & Stefan Nickel (ed.), Operations Research Proceedings 2007, pages 69-74, Springer.
    5. Tomasz Bielecki & Daniel Hernández-Hernández & Stanley R. Pliska, 1999. "Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(2), pages 167-188, October.
    6. Uriel G. Rothblum & Peter Whittle, 1982. "Growth Optimality for Branching Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 7(4), pages 582-601, November.
    7. Ronald A. Howard & James E. Matheson, 1972. "Risk-Sensitive Markov Decision Processes," Management Science, INFORMS, vol. 18(7), pages 356-369, March.
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