The Annals of Applied Probability

Risk measuring under model uncertainty

Jocelyne Bion-Nadal and Magali Kervarec

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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

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

First available in Project Euclid: 7 February 2012

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Zentralblatt MATH identifier

Primary: 46A20: Duality theory 91B30: Risk theory, insurance
Secondary: 46E05: Lattices of continuous, differentiable or analytic functions

Risk measure duality theory uncertainty capacity


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

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