Eliciting reference measures of law-invariant functionals

TL;DR

This paper introduces a method to recover or test the reference measure underlying law-invariant functionals in risk management, using observed functional values and dual space support sets, with applications to popular risk measures.

Contribution

It proposes a novel approach to identify or test the reference measure for law-invariant functionals based on dual space support sets, extending the understanding of risk measure elicitation.

Findings

Method successfully recovers reference measures for entropic risk and Expected Shortfall.

Modified approach enables elicitation for Value-at-Risk despite initial triviality issues.

Provides a theoretical framework linking law invariance, dual spaces, and measure support sets.

Abstract

Law-invariant functionals are central to risk management and assign identical values to random prospects sharing the same distribution under an atomless reference probability measure. This measure is typically assumed fixed. Here, we adopt the reverse perspective: given only observed functional values, we aim to either recover the reference measure or identify a candidate measure to test for law invariance when that property is not {\em a priori} satisfied. Our approach is based on a key observation about law-invariant functionals defined on law-invariant domains. These functionals define lower (upper) supporting sets in dual spaces of signed measures, and the suprema (infima) of these supporting sets -- if existent -- are scalar multiples of the reference measure. In specific cases, this observation can be formulated as a sandwich theorem. We illustrate the methodology through a detailed analysis of prominent examples: the entropic risk measure, Expected Shortfall, and Value-at-Risk. For the latter, our elicitation procedure initially fails due to the triviality of supporting set extrema. We therefore develop a suitable modification.