Automatic Fatou Property of Law-invariant Risk Measures

TL;DR

This paper characterizes when law-invariant risk measures automatically possess the Fatou property on certain function spaces, linking it to the AOCEA property, with implications for classical spaces like Orlicz spaces.

Contribution

It provides a necessary and sufficient condition (AOCEA property) for law-invariant risk measures to have the Fatou property on rearrangement-invariant spaces beyond L^\infty.

Findings

AOCEA property characterizes Fatou property for risk measures.

Most classical spaces, including Orlicz spaces, satisfy AOCEA.

Risk measures admit dual representations under AOCEA.

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.