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Stability properties of Haezendonck-Goovaerts premium principles

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  • Niushan Gao
  • Cosimo Munari
  • Foivos Xanthos

Abstract

We investigate a variety of stability properties of Haezendonck-Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz function. In addition, we show that Haezendonck-Goovaerts principles satisfy the stronger Lebesgue property if and only if the reference Orlicz function fulfills the so-called $\Delta_2$ condition. We also discuss (semi)continuity properties with respect to $\Phi$-weak convergence of probability measures. In particular, we show that Haezendonck-Goovaerts principles, restricted to the corresponding Young class, are always lower semicontinuous with respect to the $\Phi$-weak convergence.

Suggested Citation

  • Niushan Gao & Cosimo Munari & Foivos Xanthos, 2019. "Stability properties of Haezendonck-Goovaerts premium principles," Papers 1909.10735, arXiv.org, revised Aug 2020.
    • Handle: RePEc:arx:papers:1909.10735
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    File URL: http://arxiv.org/pdf/1909.10735
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    References listed on IDEAS

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    1. 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.
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    10. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2012. "Beyond cash-additive risk measures: when changing the num\'{e}raire fails," Papers 1206.0478, arXiv.org, revised Feb 2014.
    11. Ahn, Jae Youn & Shyamalkumar, Nariankadu D., 2014. "Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 78-90.
    12. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
    13. Wang, Xing & Peng, Liang, 2016. "Inference for intermediate Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 231-240.
    14. Tang, Qihe & Yang, Fan, 2012. "On the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 217-227.
    15. Mao, Tiantian & Hu, Taizhong, 2012. "Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 333-343.
    16. Liu, Qing & Peng, Liang & Wang, Xing, 2017. "Haezendonck–Goovaerts risk measure with a heavy tailed loss," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 28-47.
    17. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
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