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Risk Measures For Non‐Integrable Random Variables

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  • Freddy Delbaen

Abstract

We show that when a real‐valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables. As a result we see that a risk measure defined for, say, Cauchy‐distributed random variable, must take infinite values for some of the random variables.

Suggested Citation

  • Freddy Delbaen, 2009. "Risk Measures For Non‐Integrable Random Variables," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 329-333, April.
    • Handle: RePEc:bla:mathfi:v:19:y:2009:i:2:p:329-333
    DOI: 10.1111/j.1467-9965.2009.00370.x
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/342 is not listed on IDEAS
    2. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
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    Citations

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    Cited by:

    1. Assa, Hirbod & Zimper, Alexander, 2018. "Preferences over all random variables: Incompatibility of convexity and continuity," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 71-83.
    2. Gardes, Laurent, 2025. "On tail-risk measures for non-integrable heavy-tailed random variables," Econometrics and Statistics, Elsevier, vol. 35(C), pages 84-100.
    3. Niushan Gao & Foivos Xanthos, 2016. "Option spanning beyond $L_p$-models," Papers 1603.01288, arXiv.org, revised Sep 2016.
    4. Mainik Georg & Rüschendorf Ludger, 2012. "Ordering of multivariate risk models with respect to extreme portfolio losses," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 73-106, March.
    5. Svindland Gregor, 2009. "Subgradients of law-invariant convex risk measures on L," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 169-199, December.
    6. Niushan Gao & Denny H. Leung & Foivos Xanthos, 2016. "Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures," Papers 1610.08806, arXiv.org, revised Jun 2017.
    7. Niushan Gao & Foivos Xanthos, 2015. "On the C-property and $w^*$-representations of risk measures," Papers 1511.03159, arXiv.org, revised Sep 2016.
    8. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2018. "Law-invariant functionals on general spaces of random variables," Papers 1808.00821, arXiv.org, revised Jan 2021.
    9. Keita Owari, 2012. "Maximum Lebesgue Extension Of Convex Risk Measures," CARF F-Series CARF-F-287, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    10. Niushan Gao & Denny Leung & Cosimo Munari & Foivos Xanthos, 2018. "Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Finance and Stochastics, Springer, vol. 22(2), pages 395-415, April.
    11. Bernard, Carole & Jiang, Xiao & Wang, Ruodu, 2014. "Risk aggregation with dependence uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 93-108.
    12. Niushan Gao & Denny H. Leung & Cosimo Munari & Foivos Xanthos, 2017. "Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Papers 1701.05967, arXiv.org, revised Sep 2017.
    13. Christos E. Kountzakis & Damiano Rossello, 2024. "Risk Measures’ Duality on Ordered Linear Spaces," Mathematics, MDPI, vol. 12(8), pages 1-15, April.
    14. Keita Owari, 2013. "Maximum Lebesgue Extension of Monotone Convex Functions," Papers 1304.7934, arXiv.org, revised Jan 2014.
    15. Pablo Koch-Medina & Cosimo Munari, 2024. "Qualitative robustness of utility-based risk measures," Annals of Operations Research, Springer, vol. 336(1), pages 967-980, May.
    16. Niushan Gao & Cosimo Munari, 2020. "Surplus-Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1342-1370, November.
    17. Paul Embrechts & Giovanni Puccetti & Ludger Rüschendorf & Ruodu Wang & Antonela Beleraj, 2014. "An Academic Response to Basel 3.5," Risks, MDPI, vol. 2(1), pages 1-24, February.
    18. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Jan 2026.
    19. Alexander Zimper & Hirbod Assa, 2019. "Preferences Over Rich Sets of Random Variables: Semicontinuity in Measure versus Convexity," Working Papers 201940, University of Pretoria, Department of Economics.
    20. Alejandro Balbás & Iván Blanco & José Garrido, 2014. "Measuring Risk When Expected Losses Are Unbounded," Risks, MDPI, vol. 2(4), pages 1-14, September.
    21. Marcelo Brutti Righi, 2017. "Closed spaces induced by deviation measures," Economics Bulletin, AccessEcon, vol. 37(3), pages 1781-1784.

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