Robust distortion riskmetrics over Bregman--Wasserstein balls and their intersections

Abstract

We study the worst-case values of distortion riskmetrics over Bregman--Wasserstein balls around a reference distribution. Unlike the squared $2$-Wasserstein distance, Bregman--Wasserstein divergences can penalize upward and downward deviations differently. Our main result applies to generalized distortion functions $g:[0,1]\to\R$ of bounded variation, with $g(0)=0$ and suitable endpoint regularity, which need not be monotone, normalized, or upper semicontinuous. For strictly convex generators satisfying a two-sided $p$-growth condition, we obtain a one-dimensional dual representation of the worst-case value. When a positive Lagrange multiplier makes the divergence constraint binding, we identify a unique quantile function through a least concave majorant and characterize when it attains the supremum. For absolutely continuous generalized distortion functions, we characterize the solution through an isotonic projection; we also study more general generators and shape constraints, including unimodality. Finally, we extend the analysis to intersections of Bregman--Wasserstein balls, describing models acceptable to several parties involved in decision making, and illustrate the results numerically.