Paper: arXiv 2507.13763

Authors: Felix-Benedikt Liebrich, Ruodu Wang

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for eliciting reference measures using functional analysis and supporting sets, solving an interesting reverse problem in risk management. However, it is purely mathematical with no empirical data analysis, backtests, or code provided.

Research Flowchart

  flowchart TD
    A["<b>Research Goal:</b> Elicit unknown reference measure <br>from observed values of law-invariant functionals"] --> B["<b>Core Theoretical Input:</b><br>Observed functional values on law-invariant domains"]
    
    B --> C["<b>Methodology:</b><br>Construct lower/upper supporting sets <br>in dual spaces of signed measures"]
    
    C --> D["<b>Computational Process:</b><br>Calculate extrema (suprema/infima) <br>of supporting sets via Sandwich Theorem"]
    
    D --> E{"Evaluation on <br>Prominent Examples"}
    
    E -- "Standard Extrema Success" --> F["<b>Outcome 1:</b> Exact recovery of reference measure <br><i>(Entropic Risk Measure & Expected Shortfall)</i>"]
    
    E -- "Trivial Extrema Failure" --> G["<b>Outcome 2:</b> Modified elicitation procedure <br>developed to successfully recover measure <br><i>(Value-at-Risk)</i>"]