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February 2012 Risk measuring under model uncertainty
Jocelyne Bion-Nadal, Magali Kervarec
Ann. Appl. Probab. 22(1): 213-238 (February 2012). DOI: 10.1214/11-AAP766

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 Cb(Ω), we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless nonpositive elements of Cb(Ω). 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 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 E(|f|)=0. We also apply our results to the case of uncertain volatility.

Citation

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Jocelyne Bion-Nadal. Magali Kervarec. "Risk measuring under model uncertainty." Ann. Appl. Probab. 22 (1) 213 - 238, February 2012. https://doi.org/10.1214/11-AAP766

Information

Published: February 2012
First available in Project Euclid: 7 February 2012

zbMATH: 1242.46006
MathSciNet: MR2932546
Digital Object Identifier: 10.1214/11-AAP766

Subjects:
Primary: 46A20 , 91B30
Secondary: 46E05

Keywords: capacity , duality theory , risk measure , uncertainty

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.22 • No. 1 • February 2012
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