Distributions or Quantiles? A Convex Variational Theory of Worst-Case Risk

Worst-case risk under model uncertainty is central to robust portfolio choice, but closed-form results have been scattered across ambiguity sets of special shape. Gao, Li, Wang and Zhang take spectral risk measures as the base class and propose a representation principle: work with whichever formulation, quantile or distribution, keeps the whole ambiguity set convex. The resulting optimality conditions identify the worst-case law and its risk value.

The approach recovers classical central-moment bounds and gives closed or semi-closed forms for Wasserstein ambiguity with tail stress, two-sided expected-shortfall bands, asymmetric partial moments and Kullback-Leibler neighbourhoods of exponential or Gaussian references. For mean-anchored joint Kullback-Leibler ambiguity around a Gaussian, a single worst-case coefficient applies to every portfolio, so robust selection becomes an exact second-order cone program after one calibration.