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Uniqueness of Kusuoka Representations


  • Alois Pichler
  • Alexander Shapiro


This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka representation is derived from this initial result. Further, stochastic order relations are employed to identify the minimal Kusuoka representation. It is shown that measures in the minimal representation are extremal with respect to the order relations. The tools are finally employed to provide the minimal representation for important practical examples. Although the Kusuoka representation is usually given only for nonatomic probability spaces, this presentation closes the gap to spaces with atoms.

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  • Alois Pichler & Alexander Shapiro, 2012. "Uniqueness of Kusuoka Representations," Papers 1210.7257,, revised Feb 2013.
  • Handle: RePEc:arx:papers:1210.7257

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    References listed on IDEAS

    1. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2010. "Kusuoka representation of higher order dual risk measures," Annals of Operations Research, Springer, vol. 181(1), pages 325-335, December.
    2. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Convex Risk Functions," Risk and Insurance 0404001, EconWPA, revised 08 Oct 2005.
    3. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
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    Cited by:

    1. Johanna F. Ziegel, 2013. "Coherence and elicitability," Papers 1303.1690,, revised Mar 2014.
    2. Kerem Ugurlu, 2014. "On the Coherent Risk Measure Representations in the Discrete Probability Spaces," Papers 1411.4441,, revised Dec 2014.
    3. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    4. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.

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