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A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations

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  • Lesedi Mabitsela
  • Calisto Guambe
  • Rodwell Kufakunesu

Abstract

In this paper, we provide a representation theorem for dynamic capital allocation under It{\^o}-L{\'e}vy model. We consider the representation of dynamic risk measures defined under Backward Stochastic Differential Equations (BSDE) with generators that grow quadratic-exponentially in the control variables. Dynamic capital allocation is derived from the differentiability of BSDEs with jumps. The results are illustrated by deriving a capital allocation representation for dynamic entropic risk measure and static coherent risk measure.

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  • Lesedi Mabitsela & Calisto Guambe & Rodwell Kufakunesu, 2018. "A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations," Papers 1808.04611, arXiv.org.
    • Handle: RePEc:arx:papers:1808.04611
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    References listed on IDEAS

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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Yuhong Xu, 2016. "MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g-EXPECTATION," Mathematical Finance, Wiley Blackwell, vol. 26(3), pages 638-673, July.
    3. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    4. 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.
    5. Buch, A. & Dorfleitner, G., 2008. "Coherent risk measures, coherent capital allocations and the gradient allocation principle," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 235-242, February.
    6. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    7. Michael Kalkbrener, 2005. "An Axiomatic Approach To Capital Allocation," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 425-437, July.
    8. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability," Papers 1512.05924, arXiv.org, revised Sep 2017.
    9. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    10. N. C. Framstad & B. Øksendal & A. Sulem, 2004. "Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 77-98, April.
    11. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    12. Alexander S. Cherny, 2009. "Capital Allocation And Risk Contribution With Discrete‐Time Coherent Risk," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 13-40, January.
    13. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    14. Eduard Kromer & Ludger Overbeck, 2014. "Representation Of Bsde-Based Dynamic Risk Measures And Dynamic Capital Allocations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(05), pages 1-16.
    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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