Law Invariant Risk Measures Have the Fatou Property

Elyès Jouini (jouini@ceremade.dauphine.fr), Walter Schachermayer and Nizar Touzi
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Walter Schachermayer: TU Wien - Vienna University of Technology = Technische Universität Wien
Nizar Touzi: CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique

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Abstract: S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterizationof law invariant coherent risk measures, satisfying the Fatou property.The latter property was introduced by F. Delbaen [D 02]. In thepresent note we extend Kusuoka's characterization in two directions, thefirst one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. F¨ollmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.We also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property.

Keywords: Fatou property; risk measures (search for similar items in EconPapers)
Date: 2006-01-01
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00176522v1
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Citations: View citations in EconPapers (132)

Published in Advances in mathematical economics, 2006, pp.49-71

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