Characterization of optimal risk allocations for convex risk functionals

In this paper we consider the problem of optimal risk allocation or risk exchange with respect to convex risk functionals, which not necessarily are monotone or cash invariant. General existence and characterization results are given for optimal risk allocations minimizing the total risk as well as for Pareto optimal allocations. We establish a general uniqueness result for optimal allocations. As particular consequence we obtain in case of cash invariant, strictly convex risk functionals the uniqueness of Pareto optimal allocations up to additive constants. In the final part some tools are developed useful for the verification of the basic intersection condition made in the theorems which are applied in several examples.