Distribution-invariant risk measures, information, and dynamic consistency

In the first part of the paper, we characterize distribution-invariant risk measures with convex acceptance and rejection sets on the level of distributions. It is shown that these risk measures are closely related to utility-based shortfall risk. In the second part of the paper, we provide an axiomatic characterization for distribution-invariant dynamic risk measures of terminal payments. We prove a representation theorem and investigate the relation to static risk measures. A key insight of the paper is that dynamic consistency and the notion of "measure convex sets of probability measures" are intimately related. This result implies that under weak conditions dynamically consistent dynamic risk measures can be represented by static utility-based shortfall risk.