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Minimizing Spectral Risk Measures Applied to Markov Decision Processes

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  • Nicole Bauerle
  • Alexander Glauner

Abstract

We study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in B\"auerle and Ott (2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

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  • Nicole Bauerle & Alexander Glauner, 2020. "Minimizing Spectral Risk Measures Applied to Markov Decision Processes," Papers 2012.04521, arXiv.org.
    • Handle: RePEc:arx:papers:2012.04521
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    References listed on IDEAS

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    1. Georg Ch. Pflug & Alois Pichler, 2016. "Time-Consistent Decisions and Temporal Decomposition of Coherent Risk Functionals," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 682-699, May.
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    3. Nicole Bäuerle & Ulrich Rieder, 2014. "More Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 105-120, February.
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    5. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    6. Nicole Bäuerle & Jonathan Ott, 2011. "Markov Decision Processes with Average-Value-at-Risk criteria," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 361-379, December.
    7. Cui, Wei & Yang, Jingping & Wu, Lan, 2013. "Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 74-85.
    8. Nicole Bauerle & Alexander Glauner, 2020. "Markov Decision Processes with Recursive Risk Measures," Papers 2010.07220, arXiv.org.
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    Cited by:

    1. Alexander Glauner, 2020. "Dynamic Reinsurance in Discrete Time Minimizing the Insurer's Cost of Capital," Papers 2012.09648, arXiv.org.

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