🤖 AI Summary
This study investigates robust risk measures derived from worst-case convex risk measures over uncertainty sets, aiming to characterize their continuity conditions and uncover the duality relationship between such risk measures and the underlying uncertainty sets. Drawing on tools from convex analysis, duality theory, and set-valued analysis, the work proposes two complementary dual frameworks—each relying on distinct geometric assumptions—to establish a unified set-valued dual representation. The main contributions include precise criteria for the continuity of robust risk measures and dual characterizations of both the risk measures and the uncertainty sets, thereby systematically elucidating the deep interplay between them.
📝 Abstract
We consider robust risk measures that arise as worst-case values of convex risk measures evaluated on uncertainty sets. We characterize continuity properties of robust risk measures through their consolidated uncertainty sets, derive dual representations for robust risk measures, and develop a set-valued dual representation for consolidated uncertainty sets. The two dual frameworks rely on distinct geometric assumptions and are therefore complementary rather than interchangeable.