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Automatic Fatou Property of Law-invariant Risk Measures

Author

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  • Shengzhong Chen
  • Niushan Gao
  • Denny Leung
  • Lei Li
Abstract
In the paper we investigate automatic Fatou property of law-invariant risk measures on a rearrangement-invariant function space $\mathcal{X}$ other than $L^\infty$. The main result is the following characterization: Every real-valued, law-invariant, coherent risk measure on $\mathcal{X}$ has the Fatou property at every random variable $X\in \mathcal{X}$ whose negative tails have vanishing norm (i.e., $\lim_n\|X\mathbf{1}_{\{X\leq -n\}}\|=0$) if and only if $\mathcal{X}$ satisfies the Almost Order Continuous Equidistributional Average (AOCEA) property, namely, $\mathrm{d}(\mathcal{CL}(X),\mathcal{X}_a) =0$ for any $X\in \mathcal{X}_+$, where $ \mathcal{CL}(X)$ is the convex hull of all random variables having the same distribution as $X$ and $\mathcal{X}_a=\{X\in\mathcal{X}:\lim_n \|X\mathbf{1}_{ \{|X|\geq n\} }\| =0\}$. As a consequence, we show that under the AOCEA property, every real-valued, law-invariant, coherent risk measure on $\mathcal{X}$ admits a tractable dual representation at every $X\in \mathcal{X}$ whose negative tails have vanishing norm. Furthermore, we show that the AOCEA property is satisfied by most classical model spaces, including Orlicz spaces, and therefore the foregoing results have wide applications.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2107.08109
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    References listed on IDEAS

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