A Unifying Perspective on the Collapse to the Mean for Law-Invariant Functionals

Battle Count: 75

The paper is primarily relevant to risk management and regulatory capital rather than direct trading strategies. However, understanding when law-invariant risk functionals collapse to the mean is crucial for quantitative risk managers who design or calibrate risk measures for trading desks. The results inform the design of risk measures that genuinely capture tail risk and distributional features beyond the mean, which is essential for portfolio risk assessment, position sizing, and regulatory compliance. The connections to Expected Shortfall, consistent risk measures, and sensitivity to large losses are directly relevant to quantitative risk management in trading contexts.

This is a highly theoretical mathematics paper requiring advanced knowledge of functional analysis, measure theory, convex analysis, stochastic orders, and mathematical finance. There is no code to implement. Understanding and applying the results requires deep mathematical sophistication. The proofs involve intricate constructions (conditional expectations along finite partitions, two-dimensional reductions, Fatou property arguments). For practitioners, the main value is in understanding the structural conditions under which risk functionals lose distributional information, rather than in any computational implementation.

As a purely theoretical mathematics paper with complete proofs, the results are fully reproducible by any reader with the requisite mathematical background. All assumptions, lemmas, and theorems are stated precisely with full proofs provided in Section 4. No computational experiments or code are involved. The mathematical framework is self-contained with clear definitions.