Eliciting reference measures of law-invariant functionals

TL;DR

The article develops a dual, model-agnostic framework to elicit the reference probability measure underlying a law-invariant functional, by analyzing lower and upper supporting sets in the dual spaces of signed measures. It proves that, under an atomless countably additive reference, the supremum (infimum) of these supporting sets, when existing, equals a scalar multiple of the reference measure, thereby allowing identification of the reference measure and providing a test for law invariance when the property is not assumed. The framework is instantiated through distortion riskmetrics and prominent examples—Entropic risk, Expected Shortfall, and Value-at-Risk—highlighting where the dual approach succeeds or requires modification (notably for VaR). The work bridges functional-analytic duality with risk-measure theory, offering a theoretical pathway to test law invariance and recover reference measures, with potential implications for regulatory risk assessment and decision-theoretic elicitation.

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.

Theorems & Definitions (34)

• Theorem 3.1
• Example 3.2
• Example 3.3
• Corollary 3.4
• Example 3.5
• Theorem 3.6
• Lemma 3.7
• proof
• proof : Proof of Theorem \ref{['main1']}
• proof : Proof of Corollary \ref{['cor:remark']}