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Law Invariant Risk Measures Have the Fatou Property

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  • Elyès Jouini

    ()
    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - CNRS : UMR7534 - Université Paris Dauphine - Paris IX)

  • Walter Schachermayer

    (VUT - Vienna University of Technology - Technische Universität Wien)

  • Nizar Touzi

    (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique)

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.

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Bibliographic Info

Paper provided by HAL in its series Post-Print with number halshs-00176522.

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Date of creation: 01 Jan 2006
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Publication status: Published, Advances in mathematical economics, 2006, 49-71
Handle: RePEc:hal:journl:halshs-00176522

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Keywords: Fatou property; risk measures;

References

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  1. Clotilde Napp & Elyès Jouini, 2005. "Conditional Comonotonicity," Post-Print, HAL halshs-00151516, HAL.
  2. Volker Krätschmer, 2005. "Robust representation of convex risk measures by probability measures," Finance and Stochastics, Springer, Springer, vol. 9(4), pages 597-608, October.
  3. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2007. "Optimal Risk Sharing for Law Invariant Monetary Utility Functions," Working Papers halshs-00176606, HAL.
  4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
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