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A supermartingale relation for multivariate risk measures

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  • Zachary Feinstein
  • Birgit Rudloff

Abstract

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.

Suggested Citation

  • Zachary Feinstein & Birgit Rudloff, 2018. "A supermartingale relation for multivariate risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 18(12), pages 1971-1990, December.
    • Handle: RePEc:taf:quantf:v:18:y:2018:i:12:p:1971-1990
    DOI: 10.1080/14697688.2018.1459810
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    Citations

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    Cited by:

    1. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Feb 2021.
    2. Yanhong Chen & Zachary Feinstein, 2022. "Set-valued dynamic risk measures for processes and for vectors," Finance and Stochastics, Springer, vol. 26(3), pages 505-533, July.
    3. E. Kromer & L. Overbeck & K. Zilch, 2019. "Dynamic systemic risk measures for bounded discrete time processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 77-108, August.
    4. Yanhong Chen & Zachary Feinstein, 2021. "Set-Valued Dynamic Risk Measures for Processes and Vectors," Papers 2103.00905, arXiv.org, revised Nov 2021.
    5. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978, arXiv.org, revised Nov 2021.
    6. 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.
    7. 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.

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