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A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle

Author

Listed:
  • Zachary Feinstein

    (Washington University in St. Louis)

  • Birgit Rudloff

    (Institute for Statistics and Mathematics)

Abstract

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for d assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman’s principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.

Suggested Citation

  • Zachary Feinstein & Birgit Rudloff, 2017. "A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle," Journal of Global Optimization, Springer, vol. 68(1), pages 47-69, May.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:1:d:10.1007_s10898-016-0459-8
    DOI: 10.1007/s10898-016-0459-8
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Gabriela Kov'av{c}ov'a & Birgit Rudloff, 2018. "Time consistency of the mean-risk problem," Papers 1806.10981, arXiv.org, revised Jan 2020.
    2. Zachary Feinstein & Birgit Rudloff, 2024. "Deep learning the efficient frontier of convex vector optimization problems," Journal of Global Optimization, Springer, vol. 90(2), pages 429-458, October.
    3. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Feb 2021.
    4. Igor Cialenco & Gabriela Kov'av{c}ov'a, 2024. "Vector-valued robust stochastic control," Papers 2407.00266, arXiv.org.
    5. Roozegar, Roohollah & Balakrishnan, Narayanaswamy & Jamalizadeh, Ahad, 2020. "On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    6. Shushi, Tomer, 2018. "Stein’s lemma for truncated elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 297-303.
    7. Zachary Feinstein & Birgit Rudloff, 2022. "Deep Learning the Efficient Frontier of Convex Vector Optimization Problems," Papers 2205.07077, arXiv.org, revised May 2024.
    8. Wissam AlAli & c{C}au{g}{i}n Ararat, 2024. "Systemic values-at-risk and their sample-average approximations," Papers 2408.08511, arXiv.org.
    9. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978, arXiv.org, revised Nov 2021.
    10. c{C}au{g}{i}n Ararat & Zachary Feinstein, 2019. "Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations," Papers 1912.06916, arXiv.org, revised Sep 2020.
    11. Çağın Ararat & Firdevs Ulus & Muhammad Umer, 2022. "A Norm Minimization-Based Convex Vector Optimization Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 681-712, August.
    12. Çağın Ararat & Zachary Feinstein, 2021. "Set-valued risk measures as backward stochastic difference inclusions and equations," Finance and Stochastics, Springer, vol. 25(1), pages 43-76, January.
    13. Zachary Feinstein & Andreas Sojmark, 2022. "Endogenous distress contagion in a dynamic interbank model: how possible future losses may spell doom today," Papers 2211.15431, arXiv.org, revised Aug 2024.

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