🤖 AI Summary
This paper studies robust convex risk measures under distributional uncertainty, focusing on three canonical ambiguity sets: $p$-norm balls, Wasserstein balls, and mean-variance moment constraints. Leveraging convex analysis and duality theory, it derives— for the first time—the exact analytical forms of the corresponding convex conjugate penalty functions and obtains closed-form expressions for the robust risk measures over each ambiguity set. The main contributions are: (1) a unified analytical dual representation framework that characterizes the structural properties of worst-case risk; (2) simultaneous theoretical tractability and computational feasibility, substantially improving the efficiency of robust risk evaluation; and (3) a rigorous, implementable theoretical foundation for both Wasserstein-based robust optimization and moment-based robust models.
📝 Abstract
We study robust convex risk measures related to worst-case values under uncertainty in random variables. Our first main result characterizes the convex conjugate penalty term, which is the key to dual representations. Our second main result uses such penalty term to provide closed forms when uncertainty sets are based on closed balls under p-norms and Wasserstein distance.