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Law Invariant Risk Measures Have the Fatou Property

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Author Info
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)

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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.

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Paper provided by HAL in its series Post-Print with number halshs-00176522_v1.

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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_v1

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Keywords: Fatou property; risk measures;

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References listed on IDEAS
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  1. Volker Krätschmer, 2005. "Robust representation of convex risk measures by probability measures," Finance and Stochastics, Springer, vol. 9(4), pages 597-608, October. [Downloadable!] (restricted)
  2. Elyès Jouini & Clotilde Napp, 2004. "Conditional comonotonicity," Decisions in Economics and Finance, Springer, vol. 27(2), pages 153-166, December. [Downloadable!] (restricted)
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  3. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2007. "Optimal Risk Sharing for Law Invariant Monetary Utility Functions," Working Papers halshs-00176606_v1, HAL. [Downloadable!]
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Alfred Galichon, 2009. "The var at risk," Working Papers hal-00401793_v1, HAL. [Downloadable!]
  2. Beatrice Acciaio, 2009. "Short note on inf-convolution preserving the Fatou property," Annals of Finance, Springer, vol. 5(2), pages 281-287, March. [Downloadable!] (restricted)
  3. Mark Owen & Gordan Zitkovic, 2007. "Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing," Quantitative Finance Papers 0706.0478, arXiv.org, revised Sep 2007. [Downloadable!]
  4. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828_v1, HAL. [Downloadable!]
  5. Alexander Schied, 2005. "Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach," SFB 649 Discussion Papers SFB649DP2005-051, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Aug 2006. [Downloadable!]
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  6. Volker Krätschmer, 2007. "On s-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model," SFB 649 Discussion Papers SFB649DP2007-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany. [Downloadable!]
  7. Beatrice Acciaio, 2007. "Optimal risk sharing with non-monotone monetary functionals," Finance and Stochastics, Springer, vol. 11(2), pages 267-289, April. [Downloadable!] (restricted)
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