Robust distortion riskmetrics under Wasserstein ambiguity

Risk evaluation under distributional ambiguity is central to decision making in finance, economics, and operations research. Wasserstein balls provide a natural way to describe uncertainty around a reference distribution. We solve a natural yet open problem of robust optimization for the class of distortion riskmetrics with Wasserstein distance as the sole ambiguity constraint. This chosen objective class does not require convexity, monotonicity, and continuity of distortion functions, encompassing many common risk measures and deviation measures. First, we characterize conditions under which direct convexification preserves the worst-case value. Second, we develop a constructive method for exact worst-case evaluation when the direct convexification conditions fail. Third, we construct explicit approximate worst-case distributions and provide computable error bounds to assess their accuracy without solving the exact problem. We apply these results to distributionally robust portfolio selection and use numerical experiments to assess approximation accuracy and the resulting portfolio decisions.