Stochastic Finance, 4th Edition

4.6Concave distortions

Let us now have a closer look at the coherent risk measures

which appear in the Representation Theorem 4.62 for law-invariant convex risk measures. We are going to characterize these risk measures ρμ in two ways, first as Choquet integrals with respect to some concave distortion of the underlying probability measure P, and then, in the next section, by a property of comonotonicity.

Again, we will assume throughout this section that the underlying probability space (Ω,F, P) is atomless. Since AV@Rλ is coherent, continuous from below, and law-invariant, any mixture ρμ for some probability measure μ on (0, 1] has the same properties. ...