ASYMPTOTIC BEHAVIOUR OF UNEXPECTED LOSSES AND RISK RATIOS FOR CO-MONOTONIC ALTERNATIVES

Battle Count: 58

The paper is primarily relevant to risk management, insurance pricing, and regulatory capital rather than quantitative trading strategies. However, the results on risk aggregation under dependence uncertainty, worst-case bounds, and the comparison between diversified and co-monotonic portfolios have indirect relevance for portfolio risk assessment in trading contexts. The VaR and Expected Shortfall analysis is applicable to trading book risk. The theoretical framework for model risk in dependence structures is relevant for stress testing and scenario analysis in trading desks.

The paper is highly theoretical with no code or implementation guidance. Understanding requires advanced knowledge of functional analysis (Banach lattices, Orlicz spaces, Luxemburg norms), measure theory (uniform integrability, convergence in probability), probability theory (weak law of large numbers, triangular arrays), and risk theory (coherent/convex risk measures, Choquet integrals, distortion functions). The mathematical machinery (Theorem 5.4, Corollary 5.5, Namioka-Klee extensions) is sophisticated. No practical implementation is provided or implied.

This is a purely theoretical mathematical paper with complete proofs provided for all theorems, propositions, and corollaries. All definitions, assumptions, and conditions are explicitly stated. The mathematical framework (Orlicz spaces, Banach lattices, weak law of large numbers, uniform integrability) is well-established in the literature. No empirical data or code is required for verification. All results are self-contained with detailed proofs in Sections 2-5.