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Higher order measures of risk and stochastic dominance

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  • Alois Pichler

Abstract

Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained.

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  • Alois Pichler, 2024. "Higher order measures of risk and stochastic dominance," Papers 2402.15387, arXiv.org.
    • Handle: RePEc:arx:papers:2402.15387
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    References listed on IDEAS

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    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. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    3. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    4. Tom Erik Sønsteng Henriksen & Alois Pichler & Sjur Westgaard & Stein Frydenberg, 2019. "Can commodities dominate stock and bond portfolios?," Annals of Operations Research, Springer, vol. 282(1), pages 155-177, November.
    5. Alejandro Balbás & Beatriz Balbás & Raquel Balbás & Jean-Philippe Charron, 2023. "Bidual Representation of Expectiles," Risks, MDPI, vol. 11(12), pages 1-21, December.
    6. Fabio Bellini & Camilla Caperdoni, 2007. "Coherent Distortion Risk Measures and Higher-Order Stochastic Dominances," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 35-42.
    7. Ogryczak, Wlodzimierz & Ruszczynski, Andrzej, 1999. "From stochastic dominance to mean-risk models: Semideviations as risk measures," European Journal of Operational Research, Elsevier, vol. 116(1), pages 33-50, July.
    8. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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