Law Invariant Risk Measures Have the Fatou Property
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.
|Date of creation:||01 Jan 2006|
|Date of revision:|
|Publication status:||Published in Advances in mathematical economics, 2006, pp.49-71|
|Note:||View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00176522|
|Contact details of provider:|| Web page: https://hal.archives-ouvertes.fr/|
References listed on IDEAS
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.:
- Elyès Jouini & Clotilde Napp, 2004.
Decisions in Economics and Finance,
Springer, vol. 27(2), pages 153-166, December.
- Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2007.
"Optimal Risk Sharing for Law Invariant Monetary Utility Functions,"
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- Volker Krätschmer, 2005. "Robust representation of convex risk measures by probability measures," Finance and Stochastics, Springer, vol. 9(4), pages 597-608, October.
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