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# Risk measuring under model uncertainty

## Author

Listed:
• 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
as

File URL: http://arxiv.org/pdf/1004.5524

## 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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## Citations

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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.

### NEP fields

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