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Risk measuring under model uncertainty

Author

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  • Jocelyne Bion-Nadal
  • Magali Kervarec

Abstract

The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on ${\cal C}_b(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of ${\cal C}_b(\Omega)$. We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class. To get these results we study the topological properties of the dual of the Banach space $L^1(c)$ associated to a capacity $c$. As application we obtain that every $G$-expectation $\E$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure $P$ such that $P(|f|)=0$ iff $\E(|f|)=0$. We also apply our results to the case of uncertain volatility.

Suggested Citation

  • Jocelyne Bion-Nadal & Magali Kervarec, 2010. "Risk measuring under model uncertainty," Papers 1004.5524, arXiv.org, revised Dec 2010.
  • Handle: RePEc:arx:papers:1004.5524
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    File URL: http://arxiv.org/pdf/1004.5524
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    References listed on IDEAS

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    1. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    2. Freddy Delbaen & Shige Peng & Emanuela Rosazza Gianin, 2008. "Representation of the penalty term of dynamic concave utilities," Papers 0802.1121, arXiv.org, revised Dec 2009.
    3. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Roorda, Berend & Schumacher, J.M., 2007. "Time consistency conditions for acceptability measures, with an application to Tail Value at Risk," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 209-230, March.
    5. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
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    Cited by:

    1. Romain Blanchard & Laurence Carassus, 2017. "Convergence of utility indifference prices to the superreplication price in a multiple-priors framework," Papers 1709.09465, arXiv.org.

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