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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. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    2. repec:dau:papers:123456789/342 is not listed on IDEAS
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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. Niushan Gao & Foivos Xanthos, 2016. "Option spanning beyond $L_p$-models," Papers 1603.01288, arXiv.org, revised Sep 2016.
    3. 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.
    4. 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.
    5. Niushan Gao & Foivos Xanthos, 2015. "On the C-property and $w^*$-representations of risk measures," Papers 1511.03159, arXiv.org, revised Sep 2016.
    6. 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.
    7. 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.
    8. Bernard, Carole & Jiang, Xiao & Wang, Ruodu, 2014. "Risk aggregation with dependence uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 93-108.
    9. 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.
    10. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    11. Keita Owari, 2013. "Maximum Lebesgue Extension of Monotone Convex Functions," CARF F-Series CARF-F-315, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    12. Marcelo Brutti Righi, 2017. "Closed spaces induced by deviation measures," Economics Bulletin, AccessEcon, vol. 37(3), pages 1781-1784.
    13. Svindland Gregor, 2009. "Subgradients of law-invariant convex risk measures on L," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 169-199, December.
    14. 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.
    15. Christos E. Kountzakis & Damiano Rossello, 2024. "Risk Measures’ Duality on Ordered Linear Spaces," Mathematics, MDPI, vol. 12(8), pages 1-15, April.
    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. 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.
    19. Alejandro Balbás & Iván Blanco & José Garrido, 2014. "Measuring Risk When Expected Losses Are Unbounded," Risks, MDPI, vol. 2(4), pages 1-14, September.

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