Extremal Mean-Variance Functionals over Wasserstein Balls: Applications to Risk Sharing

We characterize the worst- and best-case values of a mean-variance functional over a 2-Wasserstein ball. Using quantile representations and the geometry of attainable means and standard deviations, we reduce both infinite-dimensional problems to scalar equations and construct the extremal laws as location-scale transformations of the reference distribution. Their values depend on the reference law only through its first two moments. We then derive dual representations and formulate proportional risk sharing under heterogeneous beliefs as a finite-dimensional optimization problem. Under homogeneous beliefs, we show that the classical proportional allocation remains optimal for every ambiguity radius and $\alpha$-maxmin weight. Finally, we study a coupled distortion-variance functional and characterize its worst-case quantile through a convex-envelope construction, allowing the extremal law to change in shape as well as location and scale.