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Statistics & Risk Modeling

with Applications in Finance and Insurance

Editor-in-Chief: Stelzer, Robert

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Volume 28, Issue 3


Law invariant risk measures on L (ℝd)

Ivar Ekeland / Walter Schachermayer
Published Online: 2011-11-04 | DOI: https://doi.org/10.1524/stnd.2011.1099


Kusuoka (2001) has obtained explicit representation theorems for comonotone risk measures and, more generally, for law invariant risk measures. These theorems pertain, like most of the previous literature, to the case of scalar-valued risks.

Jouini, Meddeb, and Touzi (2004) and Burgert and Rüschendorf (2006) extended the notion of risk measures to the vector-valued case. Recently Ekeland, Galichon, and Henry (2009) and Rüschendorf (2006, 2010) obtained extensions of the above theorems of Kusuoka to this setting. Their results were confined to the regular case.

In general, Kusuoka´s representation theorem for comonotone risk measures also involves a singular part. In the present work we give a full generalization of Kusuoka´s theorems to the vector-valued case. The singular component turns out to have a richer structure than in the scalar case.

Keywords: Monge–Kantorovich problem; Monge–Kantorovich duality; risk measures; law invariance

About the article

* Correspondence address: Vienna University of Technology, Financial and Actualrial Mathematics, Wiedner Hauptstraße 8-10, 1040 Vienna, Österreich,

Published Online: 2011-11-04

Published in Print: 2011-09-01

Citation Information: Statistics & Risk Modeling with Applications in Finance and Insurance, Volume 28, Issue 3, Pages 195–225, ISSN (Print) 2193-1402, DOI: https://doi.org/10.1524/stnd.2011.1099.

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