Applicability of Bernard et al. (2024) methods to p-Wasserstein empirical models

Determine whether the solution techniques developed by Bernard, Pesenti, and Vanduffel (2024) for robust distortion risk measures under 2-order Wasserstein balls (with or without moment constraints) can be applied to the optimization problems that maximize distortion risk measures over distributional uncertainty sets defined by (i) a general p-order Wasserstein ball centered at the empirical distribution function and (ii) a general p-order Wasserstein ball with fixed mean and pth moment centered at the empirical distribution function.

Background

The paper investigates worst-case distortion risk measures over two distributional uncertainty sets: a general p-order Wasserstein ball centered at the empirical distribution, and an augmented set with known mean and pth moment. The authors develop new methods based on Lagrangian multipliers to derive closed-form maximizers, emphasizing that their approach differs from Bernard et al. (2024), who worked with the 2-order Wasserstein distance and possibly different reference distributions.

In discussing methodological differences, the authors explicitly state uncertainty about whether Bernard et al.'s techniques could be transferred to their more general p-order Wasserstein setting centered at empirical distributions with potentially additional moment constraints, thereby posing an unresolved methodological question.