Adapted Law Invariance and Time-Consistent Dynamic Risk Measures

2026-07-05

AI Summary

This study investigates principles for dynamic financial risk measurement that depend solely on the distribution of random variables and the associated information revelation process. Addressing time-consistent dynamic risk measures, it introduces “adapted law invariance” as the dynamic counterpart to static law invariance, thereby overcoming the limitations of terminal law invariance. By employing techniques such as Fatou regularity, conditional extensions, and backward composition, the paper establishes an adapted Kusuoka representation in the coherent setting and extends the Kupper–Schachermayer theorem. The main contribution lies in proving that time-consistent risk measures satisfying adapted law invariance can be recursively generated from static law-invariant risk measures, and in fully characterizing their structural properties.

Abstract

In static risk measurement, law invariance expresses the principle that the risk of a position should depend only on its distribution, and not on the particular probability space on which it is represented. In a dynamic setting, the same principle leads naturally to adapted law invariance: the risk assessment should depend only on the probabilistic structure of the financial position together with the way information about it is revealed over time. We show that, for time-consistent risk measures, adapted law invariance is equivalent to a recursive one-step conditional-law representation. More precisely, assuming Fatou regularity, the one-step risk evaluations are exactly conditional lifts of static law-invariant risk measures, and the full dynamic risk measure is obtained by backward composition of these one-step maps. Convexity and coherence of the dynamic risk measure are characterized by the corresponding properties of the static one-step risk measures. This identifies adapted law invariance as the dynamic counterpart of ordinary law invariance. It also clarifies the strength of terminal-law invariance, as it appears in the rigidity theorem of Kupper and Schachermayer: it does not distinguish risks with the same distribution but different times of resolution. We further obtain an adapted Kusuoka representation in the coherent case and establish an extension of the Kupper--Schachermayer theorem.

Problem

• law invariance
• time-consistent dynamic risk measures
• adapted law invariance
• conditional-law representation
• risk measurement

Innovation

• adapted law invariance
• time-consistent dynamic risk measures
• conditional-law representation
• Kusuoka representation
• Fatou regularity