Variational Risk Measures

We consider a class of law-invariant convex risk measures which have a.robust representation of the form ρ(X ) = ...... . The supremum is taken over the set of all Radon Nikodym derivatives corresponding to the set of all probability measures on B(0,1] which are absolutely continuous with respect to theLebesgue measure. h is a positive and strictly decreasing function. We provide necessary and sufficient conditions for X such that ρ(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 classical AV@R.