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The natural Banach space for version independent risk measures

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  • Pichler, Alois

Abstract

Risk measures, or coherent measures of risk, are often considered on the space L∞, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure–the Average Value-at-Risk–are well defined on the larger space L1 and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some Lp space. But in many situations this is possibly unnatural, because any Lp with p>p0, say, is suitable to define the spectral risk measure as well. In addition to that, risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is?

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  • Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    • Handle: RePEc:eee:insuma:v:53:y:2013:i:2:p:405-415
    DOI: 10.1016/j.insmatheco.2013.07.005
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    Cited by:

    1. Chen Shengzhong & Gao Niushan & Xanthos Foivos, 2018. "The strong Fatou property of risk measures," Dependence Modeling, De Gruyter, vol. 6(1), pages 183-196, October.
    2. Pflug, Georg Ch. & Pichler, Alois, 2016. "Time-inconsistent multistage stochastic programs: Martingale bounds," European Journal of Operational Research, Elsevier, vol. 249(1), pages 155-163.
    3. Alois Pichler, 2024. "Connection between higher order measures of risk and stochastic dominance," Computational Management Science, Springer, vol. 21(2), pages 1-28, December.
    4. Bäuerle, Nicole & Glauner, Alexander, 2022. "Markov decision processes with recursive risk measures," European Journal of Operational Research, Elsevier, vol. 296(3), pages 953-966.
    5. Marcelo Brutti Righi, 2017. "Closed spaces induced by deviation measures," Economics Bulletin, AccessEcon, vol. 37(3), pages 1781-1784.
    6. Jonathan Yu-Meng Li, 2016. "Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization," Papers 1609.04065, arXiv.org.
    7. Koch-Medina Pablo & Munari Cosimo, 2014. "Law-invariant risk measures: Extension properties and qualitative robustness," Statistics & Risk Modeling, De Gruyter, vol. 31(3-4), pages 215-236, December.
    8. Niushan Gao & Cosimo Munari, 2020. "Surplus-Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1342-1370, November.
    9. Felix-Benedikt Liebrich & Gregor Svindland, 2018. "Risk sharing for capital requirements with multidimensional security markets," Papers 1809.10015, arXiv.org.
    10. Debora Daniela Escobar & Georg Ch. Pflug, 2020. "The distortion principle for insurance pricing: properties, identification and robustness," Annals of Operations Research, Springer, vol. 292(2), pages 771-794, September.
    11. Sebastian Fuchs & Ruben Schlotter & Klaus D. Schmidt, 2017. "A Review and Some Complements on Quantile Risk Measures and Their Domain," Risks, MDPI, vol. 5(4), pages 1-16, November.
    12. Daniela Escobar & Georg Pflug, 2018. "The distortion principle for insurance pricing: properties, identification and robustness," Papers 1809.06592, arXiv.org.
    13. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    14. Alexander Glauner, 2020. "Dynamic Reinsurance in Discrete Time Minimizing the Insurer's Cost of Capital," Papers 2012.09648, arXiv.org.
    15. Pichler, Alois & Tomasgard, Asgeir, 2016. "Nonlinear stochastic programming–With a case study in continuous switching," European Journal of Operational Research, Elsevier, vol. 252(2), pages 487-501.
    16. Felix-Benedikt Liebrich & Max Nendel, 2020. "Separability vs. robustness of Orlicz spaces: financial and economic perspectives," Papers 2009.09007, arXiv.org, revised May 2021.
    17. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    18. Shengzhong Chen & Niushan Gao & Foivos Xanthos, 2018. "The strong Fatou property of risk measures," Papers 1805.05259, arXiv.org.
    19. Georg Ch. Pflug & Alois Pichler, 2016. "Time-Consistent Decisions and Temporal Decomposition of Coherent Risk Functionals," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 682-699, May.
    20. Felix-Benedikt Liebrich & Gregor Svindland, 2019. "Risk sharing for capital requirements with multidimensional security markets," Finance and Stochastics, Springer, vol. 23(4), pages 925-973, October.
    21. Nicole Bauerle & Alexander Glauner, 2020. "Markov Decision Processes with Recursive Risk Measures," Papers 2010.07220, arXiv.org.

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