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.

References

We are not certain whether the approaches of Bernard et al. (2024) could be applied to the present models.

— On data-driven robust distortion risk measures for non-negative risks with partial information  (2508.10682 - Han et al., 14 Aug 2025) in Section 1 (Introduction)

Therefore, extending our model to accommodate divergence-based uncertainty sets would require the development of entirely new analytical tools, which is beyond the scope of the current paper. Nevertheless, this is a compelling direction for future research.

— Conditional value-at-risk under reward-penalty mechanism with applications to robust portfolio management  (2610.09246 - Cai et al., 7 Oct 2026) in Section 3, “Worst-case CVaR under reward-penalty mechanism,” discussion following equation (WC-g-l-d-M)