Representation results for law invariant time consistent functions

We show that the only dynamic risk measure which is law invariant, time consistent and relevant is the entropic one. Moreover, a real valued function c on L∞(a, b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form c(X) = u−1 ◦ E [u(X)], where u : (a, b) → R is a strictly increasing, continuous function. The proofs rely on a discrete version of the Skorohod embedding theorem.