Convex and conditionally law-invariant risk measures

Abstract

This work generalizes existing representation theorems for unconditional risk measures to the conditional case, for both static and dynamic settings. First, we derive a series of equivalent representations for conditionally law-invariant convex risk measures and their penalty functions in the static and conditional framework. These characterizations are expressed in terms of integrals of conditional quantiles, conditional average value-at-risk, random concave distortions of the conditional probability measures, and random Choquet integrals of transition capacities. Second, we delve into one-step law invariant and iterative dynamic risk measures to reconcile their time consistency and relevance by weakening law invariance. Representation theorems for this family in a finite-time setting are also explored. Then we apply the results to describe the features of a suitable class of conditionally law-invariant convex risk measures based on utility functions. Finally, we conclude by analyzing some further application of dynamic and discrete-time representations to the understanding of limiting continuous-time risk assessment as 𝑔-expectations.