On data-driven robust distortion risk measures for non-negative risks with partial information

Paper: arXiv 2508.10682

Authors: Xiangyu Han, Yijun Hu, Ran Wang, Linxiao Wei

Abstract

In this paper, by proposing two new kinds of distributional uncertainty sets, we explore robustness of distortion risk measures against distributional uncertainty. To be precise, we first consider a distributional uncertainty set which is characterized solely by a ball determined by general Wasserstein distance centered at certain empirical distribution function, and then further consider additional constraints of known first moment and any other higher moment of the underlying loss distribution function. Under the assumption that the distortion function is strictly concave and twice differentiable, and that the underlying loss random variable is non-negative and bounded, we derive closed-form expressions for the distribution functions which maximize a given distortion risk measure over the distributional uncertainty sets respectively. Moreover, we continue to study the general case of a concave distortion function and unbounded loss random variables. Comparisons with existing studies are also made. Finally, we provide a numerical study to illustrate the proposed models and results. Our work provides a novel generalization of several known achievements in the literature.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework expanding robust distortion risk measures to general p-order Wasserstein balls with higher-order moment constraints. However, it is primarily a mathematical paper with heavy real-analysis and optimization theory, offering only minimal numerical illustrations rather than empirical financial backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Robustness of Distortion Risk Measures under Distributional Uncertainty] --> B[Inputs: Non-negative Losses, Bounded/Unbounded Risks & Partial Moments]
    B --> C[Construct Uncertainty Sets: Wasserstein Ball ± Known First & Higher Moments]
    C --> D[Optimization Process: Maximize Distortion Risk Measure over Uncertainty Sets]
    D --> E[Derive Closed-Form Optimal Distribution Functions Concave Distortion Functions]
    E --> F[Outcomes: Analytical Risk Bounds, Comparison with Literature & Numerical Study]