On a class of law invariant convex risk measures

We consider the class of law invariant convex risk measures with robust representation r_h,p(X)=sup_fò₀¹ AV@R_s(X)f(s)-f^p(s)h(s)] ds\rho_{h,p}(X)=\sup_{f}\int_{0}^{1} AV@R_{s}(X)f(s)-f^{p}(s)h(s)]\,ds, where 1≤p<∞ and h is a positive and strictly decreasing function. The supremum is taken over the set of all Radon–Nikodym derivatives corresponding to the set of all probability measures on (0,1] which are absolutely continuous with respect to Lebesgue measure. We provide necessary and sufficient conditions for the position X such that ρ _h,p (X) is real-valued and the supremum is attained. Using variational methods, an explicit formula for the maximizer is given. We exhibit two examples of such risk measures and compare them to the average value at risk.