Distribution or Quantile? Convex Variational Theory for Worst-Case Risk

This study constructs a convex variational theory based on spectral risk measures, unifying the fragmented analytical conclusions of existing worst-case risk assessment and extending them to richer ambiguity models. The theory can provide closed-form or semi-closed-form solutions for multiple practical models including Wasserstein ambiguity with tail pressure and two-sided expected shortfall intervals, and can also generate projection algorithms with fixed-grid convergence. In high-dimensional portfolio applications, this theory can transform robust portfolio selection into an exact second-order cone programming problem.