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Stochastic control of optimized certainty equivalents

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  • Julio Backhoff Veraguas
  • A. Max Reppen
  • Ludovic Tangpi

Abstract

Optimized certainty equivalents (OCEs) is a family of risk measures widely used by both practitioners and academics. This is mostly due to its tractability and the fact that it encompasses important examples, including entropic risk measures and average value at risk. In this work we consider stochastic optimal control problems where the objective criterion is given by an OCE risk measure, or put in other words, a risk minimization problem for controlled diffusions. A major difficulty arises since OCEs are often time inconsistent. Nevertheless, via an enlargement of state space we achieve a substitute of sorts for time consistency in fair generality. This allows us to derive a dynamic programming principle and thus recover central results of (risk-neutral) stochastic control theory. In particular, we show that the value of our risk minimization problem can be characterized via the viscosity solution of a Hamilton--Jacobi--Bellman--Issacs equation. We further establish the uniqueness of the latter under suitable technical conditions.

Suggested Citation

  • Julio Backhoff Veraguas & A. Max Reppen & Ludovic Tangpi, 2020. "Stochastic control of optimized certainty equivalents," Papers 2001.10108, arXiv.org, revised Jun 2022.
    • Handle: RePEc:arx:papers:2001.10108
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    References listed on IDEAS

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    1. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    2. 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.
    3. Pflug, Georg Ch. & Pichler, Alois, 2016. "Time-inconsistent multistage stochastic programs: Martingale bounds," European Journal of Operational Research, Elsevier, vol. 249(1), pages 155-163.
    4. 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.
    5. Aharon Ben‐Tal & Marc Teboulle, 2007. "An Old‐New Concept Of Convex Risk Measures: The Optimized Certainty Equivalent," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 449-476, July.
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    Cited by:

    1. Gechun Liang & Yifan Sun & Thaleia Zariphopoulou, 2023. "Representation of forward performance criteria with random endowment via FBSDE and application to forward optimized certainty equivalent," Papers 2401.00103, arXiv.org.

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