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

TL;DR

This paper introduces new distributional uncertainty sets to analyze the robustness of distortion risk measures for non-negative risks, deriving closed-form solutions and exploring their properties under various conditions.

Contribution

It proposes two novel types of uncertainty sets based on Wasserstein distance and moment constraints, extending the robustness analysis of distortion risk measures.

Findings

Closed-form expressions for worst-case distribution functions.

Extension to unbounded loss variables and general concave distortion functions.

Numerical illustrations demonstrating the models' effectiveness.

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.