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In the first part of the article, 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.
We derive a continuous time approximation of the evolutionary market selection model of Blume &
Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous
time approximation equals the solution of an integral equation
in a random environment. For constant asset returns, the integral equation reduces to an autonomous
ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to
Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.