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Set-Valued Dynamic Risk Measures for Processes and Vectors

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  • Yanhong Chen
  • Zachary Feinstein

Abstract

The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast with scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilize this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multiportfolio time consistency between set-valued risk measures for processes and vectors is provided; to accomplish this, an augmented definition for multiportfolio time consistency of set-valued risk measures for processes is proposed.

Suggested Citation

  • Yanhong Chen & Zachary Feinstein, 2021. "Set-Valued Dynamic Risk Measures for Processes and Vectors," Papers 2103.00905, arXiv.org, revised Nov 2021.
  • Handle: RePEc:arx:papers:2103.00905
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    References listed on IDEAS

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    1. Marco Frittelli & Giacomo Scandolo, 2006. "Risk Measures And Capital Requirements For Processes," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 589-612, October.
    2. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    3. Feinstein Zachary & Rudloff Birgit, 2021. "Time consistency for scalar multivariate risk measures," Statistics & Risk Modeling, De Gruyter, vol. 38(3-4), pages 71-90, July.
    4. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    5. Zachary Feinstein & Birgit Rudloff, 2013. "Time consistency of dynamic risk measures in markets with transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 13(9), pages 1473-1489, September.
    6. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    7. repec:dau:papers:123456789/353 is not listed on IDEAS
    8. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561, arXiv.org, revised Jan 2018.
    9. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2005. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 9(3), pages 369-387, July.
    10. Zachary Feinstein & Birgit Rudloff, 2018. "A supermartingale relation for multivariate risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 18(12), pages 1971-1990, December.
    11. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978, arXiv.org, revised Nov 2021.
    12. Yanhong Chen & Yijun Hu, 2020. "Set-Valued Dynamic Risk Measures For Bounded Discrete-Time Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(03), pages 1-42, May.
    13. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    14. Zachary Feinstein & Birgit Rudloff, 2015. "Multi-portfolio time consistency for set-valued convex and coherent risk measures," Finance and Stochastics, Springer, vol. 19(1), pages 67-107, January.
    15. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    16. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath & Hyejin Ku, 2007. "Coherent multiperiod risk adjusted values and Bellman’s principle," Annals of Operations Research, Springer, vol. 152(1), pages 5-22, July.
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