Dual Representation of Robust Risk Measures and Uncertainty Sets

Battle Count: 80

The paper provides foundational theoretical tools for robust risk measurement under model uncertainty, which is directly relevant to quantitative trading risk management. The dual representations enable practitioners to understand worst-case risk exposure and characterize uncertainty sets. However, the paper is purely theoretical with no direct trading strategies, backtests, or empirical validation. The results are most relevant for risk managers designing robust portfolio constraints and for researchers developing distributionally robust optimization frameworks for trading.

The paper requires advanced knowledge of functional analysis (L∞ spaces, weak* topologies, Banach-Alaoglu theorem), convex analysis (Fenchel conjugation, penalty functions), set-valued analysis (Painlevé-Kuratowski limits, set-concavity/convexity), and Hamel's set-valued duality theory. Implementation would require solving infinite-dimensional optimization problems and computing support functions of set-valued maps. The theoretical results are not directly implementable without significant additional work to discretize and approximate.

As a purely theoretical mathematics paper, all results are stated as theorems, lemmas, and propositions with complete proofs. The mathematical framework is self-contained with clear definitions. Reproducibility depends on the reader's ability to verify the proofs. No computational experiments are needed. The paper builds on established results from [31] (Moresco, Mailhot, Pesenti 2025) and Hamel's set-valued duality theory [20].