Automatic Fatou Property of Law-invariant Risk Measures

Shengzhong Chen, Niushan Gao, Denny Leung and Lei Li

Papers from arXiv.org

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.

Date: 2021-07, Revised 2022-01
New Economics Papers: this item is included in nep-rmg
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://arxiv.org/pdf/2107.08109 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2107.08109

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().