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Average cost Markov control processes: stability with respect to the Kantorovich metric

Author

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  • Evgueni Gordienko
  • Enrique Lemus-Rodríguez
  • Raúl Montes-de-Oca

Abstract

We study perturbations of a discrete-time Markov control process on a general state space. The amount of perturbation is measured by means of the Kantorovich distance. We assume that an average (per unit of time on the infinite horizon) optimal control policy can be found for the perturbed (supposedly known) process, and that it is used to control the original (unperturbed) process. The one-stage cost is not assumed to be bounded. Under Lyapunov-like conditions we find upper bounds for the average cost excess when such an approximation is used in place of the optimal (unknown) control policy. As an application of the found inequalities we consider the approximation by relevant empirical distributions. We illustrate our results by estimating the stability of a simple autoregressive control process. Also examples of unstable processes are provided. Copyright Springer-Verlag 2009

Suggested Citation

  • Evgueni Gordienko & Enrique Lemus-Rodríguez & Raúl Montes-de-Oca, 2009. "Average cost Markov control processes: stability with respect to the Kantorovich metric," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(1), pages 13-33, August.
    • Handle: RePEc:spr:mathme:v:70:y:2009:i:1:p:13-33
    DOI: 10.1007/s00186-008-0229-6
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    References listed on IDEAS

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    1. Evgueni Gordienko & Enrique Lemus-Rodríguez & Raúl Montes-de-Oca, 2008. "Discounted cost optimality problem: stability with respect to weak metrics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 77-96, August.
    2. Evgueni Gordienko & Alexander Yushkevich, 2003. "Stability estimates in the problem of average optimal switching of a Markov chain," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(3), pages 345-365, August.
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    Cited by:

    1. Guan, Z. & Philpott, A.B., 2011. "A multistage stochastic programming model for the New Zealand dairy industry," International Journal of Production Economics, Elsevier, vol. 134(2), pages 289-299, December.
    2. J. Bonnans & Zhihao Cen & Thibault Christel, 2012. "Energy contracts management by stochastic programming techniques," Annals of Operations Research, Springer, vol. 200(1), pages 199-222, November.

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    1. Evgueni Gordienko & Enrique Lemus-Rodríguez & Raúl Montes-de-Oca, 2008. "Discounted cost optimality problem: stability with respect to weak metrics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 77-96, August.

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