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A note on continuity and asymptotic consistency of measures of risk and variability

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  • Niushan Gao
  • Foivos Xanthos

Abstract

In this short note, we show that every convex, order bounded above functional on a Frechet lattice is automatically norm continuous. This improves a result in \cite{RS06} and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in \cite{KSZ14}.

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  • Niushan Gao & Foivos Xanthos, 2024. "A note on continuity and asymptotic consistency of measures of risk and variability," Papers 2405.09766, arXiv.org, revised Oct 2024.
    • Handle: RePEc:arx:papers:2405.09766
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    References listed on IDEAS

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    1. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    2. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    3. Niushan Gao & Foivos Xanthos, 2018. "On the C†property and w∗†representations of risk measures," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 748-754, April.
    4. 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.
    5. Shengzhong Chen & Niushan Gao & Denny Leung & Lei Li, 2021. "Automatic Fatou Property of Law-invariant Risk Measures," Papers 2107.08109, arXiv.org, revised Jan 2022.
    6. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2017. "Domains of weak continuity of statistical functionals with a view toward robust statistics," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 1-19.
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    8. Niushan Gao & Cosimo Munari & Foivos Xanthos, 2019. "Stability properties of Haezendonck-Goovaerts premium principles," Papers 1909.10735, arXiv.org, revised Aug 2020.
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    12. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    13. Chen, Shengzhong & Gao, Niushan & Leung, Denny H. & Li, Lei, 2022. "Automatic Fatou property of law-invariant risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 105(C), pages 41-53.
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