## The Annals of Applied Probability

### Risk measuring under model uncertainty

#### 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 $\mathcal{C}_{b}(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless nonpositive elements of $\mathcal{C}_{b}(\Omega)$. 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 L1(c) associated to a capacity c.

As application, we obtain that every G-expectation $\mathbb{E}$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure P such that P(|f|) = 0 if and only iff $\mathbb{E}(|f|)=0$. We also apply our results to the case of uncertain volatility.

#### Article information

Source
Ann. Appl. Probab., Volume 22, Number 1 (2012), 213-238.

Dates
First available in Project Euclid: 7 February 2012

https://projecteuclid.org/euclid.aoap/1328623699

Digital Object Identifier
doi:10.1214/11-AAP766

Mathematical Reviews number (MathSciNet)
MR2932546

Zentralblatt MATH identifier
1242.46006

#### Citation

Bion-Nadal, Jocelyne; Kervarec, Magali. Risk measuring under model uncertainty. Ann. Appl. Probab. 22 (2012), no. 1, 213--238. doi:10.1214/11-AAP766. https://projecteuclid.org/euclid.aoap/1328623699

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