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Kusuoka representation of higher order dual risk measures

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  • Darinka Dentcheva
  • Spiridon Penev
  • Andrzej Ruszczyński

Abstract

We derive representations of higher order dual measures of risk in [InlineEquation not available: see fulltext.] spaces as suprema of integrals of Average Values at Risk with respect to probability measures on (0,1] (Kusuoka representations). The suprema are taken over convex sets of probability measures. The sets are described by constraints on the dual norms of certain transformations of distribution functions. For p=2, we obtain a special description of the set and we relate the measures of risk to the Fano factor in statistics. Copyright Springer Science+Business Media, LLC 2010

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  • 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.
    • Handle: RePEc:spr:annopr:v:181:y:2010:i:1:p:325-335:10.1007/s10479-010-0747-5
    DOI: 10.1007/s10479-010-0747-5
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    References listed on IDEAS

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    Cited by:

    1. Nilay Noyan & Gábor Rudolf, 2015. "Kusuoka representations of coherent risk measures in general probability spaces," Annals of Operations Research, Springer, vol. 229(1), pages 591-605, June.
    2. Matmoura, Yassine & Penev, Spiridon, 2013. "Multistage optimization of option portfolio using higher order coherent risk measures," European Journal of Operational Research, Elsevier, vol. 227(1), pages 190-198.
    3. Alois Pichler, 2024. "Higher order measures of risk and stochastic dominance," Papers 2402.15387, arXiv.org.
    4. Marcelo Brutti Righi, 2019. "A composition between risk and deviation measures," Annals of Operations Research, Springer, vol. 282(1), pages 299-313, November.
    5. da Costa, B. Freitas Paulo & Pesenti, Silvana M. & Targino, Rodrigo S., 2023. "Risk budgeting portfolios from simulations," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1040-1056.
    6. Massoomeh Rahsepar & Foivos Xanthos, 2020. "On the extension property of dilatation monotone risk measures," Papers 2002.11865, arXiv.org.
    7. Mao, Tiantian & Hu, Taizhong, 2012. "Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 333-343.
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    9. Pichler, Alois & Schlotter, Ruben, 2020. "Entropy based risk measures," European Journal of Operational Research, Elsevier, vol. 285(1), pages 223-236.
    10. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.
    11. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    12. Gómez, Fabio & Tang, Qihe & Tong, Zhiwei, 2022. "The gradient allocation principle based on the higher moment risk measure," Journal of Banking & Finance, Elsevier, vol. 143(C).
    13. Alois Pichler & Alexander Shapiro, 2012. "Uniqueness of Kusuoka Representations," Papers 1210.7257, arXiv.org, revised Feb 2013.
    14. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.

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