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Dual representations of quasiconvex compositions with applications to systemic risk

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  • c{C}au{g}{i}n Ararat
  • Mucahit Aygun

Abstract

Motivated by the problem of finding dual representations for quasiconvex systemic risk measures in financial mathematics, we study quasiconvex compositions in an abstract infinite-dimensional setting. We calculate an explicit formula for the penalty function of the composition in terms of the penalty functions of the ingredient functions. The proof makes use of a nonstandard minimax inequality (rather than equality as in the standard case) that is available in the literature. In the second part of the paper, we apply our results in concrete probabilistic settings for systemic risk measures, in particular, in the context of Eisenberg-Noe clearing model. We also provide novel economic interpretations of the dual representations in these settings.

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  • c{C}au{g}{i}n Ararat & Mucahit Aygun, 2021. "Dual representations of quasiconvex compositions with applications to systemic risk," Papers 2108.12910, arXiv.org.
    • Handle: RePEc:arx:papers:2108.12910
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    1. Jean-Paul Penot & Michel Volle, 1990. "On Quasi-Convex Duality," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 597-625, November.
    2. 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.
    3. Frittelli Marco & Maggis Marco, 2014. "Complete duality for quasiconvex dynamic risk measures on modules of the Lp-type," Statistics & Risk Modeling, De Gruyter, vol. 31(1), pages 103-128, March.
    4. Samuel Drapeau & Michael Kupper, 2013. "Risk Preferences and Their Robust Representation," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 28-62, February.
    5. Francesca Biagini & Jean‐Pierre Fouque & Marco Frittelli & Thilo Meyer‐Brandis, 2019. "A unified approach to systemic risk measures via acceptance sets," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 329-367, January.
    6. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
    7. Elisa Mastrogiacomo & Emanuela Rosazza Gianin, 2015. "Portfolio Optimization with Quasiconvex Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1042-1059, October.
    8. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, February.
    9. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Complete Monotone Quasiconcave Duality," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 321-339, May.
    10. Chen Chen & Garud Iyengar & Ciamac C. Moallemi, 2013. "An Axiomatic Approach to Systemic Risk," Management Science, INFORMS, vol. 59(6), pages 1373-1388, June.
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    Cited by:

    1. c{C}au{g}{i}n Ararat & Bar{i}c{s} Bilir & Elisa Mastrogiacomo, 2022. "Decomposable sums and their implications on naturally quasiconvex risk measures," Papers 2201.05686, arXiv.org.

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