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First-order sensitivity of the optimal value in a Markov decision model with respect to deviations in the transition probability function

Author

Listed:
  • Patrick Kern

    (Saarland University)

  • Axel Simroth

    (Fraunhofer Institute for Transportation and Infrastructure Systems)

  • Henryk Zähle

    (Saarland University)

Abstract

Markov decision models (MDM) used in practical applications are most often less complex than the underlying ‘true’ MDM. The reduction of model complexity is performed for several reasons. However, it is obviously of interest to know what kind of model reduction is reasonable (in regard to the optimal value) and what kind is not. In this article we propose a way how to address this question. We introduce a sort of derivative of the optimal value as a function of the transition probabilities, which can be used to measure the (first-order) sensitivity of the optimal value w.r.t. changes in the transition probabilities. ‘Differentiability’ is obtained for a fairly broad class of MDMs, and the ‘derivative’ is specified explicitly. Our theoretical findings are illustrated by means of optimization problems in inventory control and mathematical finance.

Suggested Citation

  • Patrick Kern & Axel Simroth & Henryk Zähle, 2020. "First-order sensitivity of the optimal value in a Markov decision model with respect to deviations in the transition probability function," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 165-197, August.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00706-w
    DOI: 10.1007/s00186-020-00706-w
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    References listed on IDEAS

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    1. Alfred Müller, 1997. "How Does the Value Function of a Markov Decision Process Depend on the Transition Probabilities?," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 872-885, November.
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    4. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    5. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    6. K. Hinderer, 2005. "Lipschitz Continuity of Value Functions in Markovian Decision Processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 3-22, September.
    7. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
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    Cited by:

    1. Henryk Zähle, 2022. "A concept of copula robustness and its applications in quantitative risk management," Finance and Stochastics, Springer, vol. 26(4), pages 825-875, October.

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