Characterization of optimal risk allocations for convex risk functionals Swen Kiesel and Ludger R¨ uschendorf University of Freiburg Abstract 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 also 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 devel- oped useful for the verification of the basic intersection condition made in the theorems which are applied in several examples. 1 Introduction In this paper we consider the optimal risk allocation resp. risk exchange problem de- fined as follows. Let(Ω, A,P ) be a probability space and let  i : L ∞ (P ) → (-∞, ∞], 1 ≤ i ≤ n, be convex, normed (i.e.  i (0) = 0), lower semicontinuous (lsc) risk func- tionals, describing the risk evaluation of n traders in the market. For X ∈ L ∞ define A(X ) :=  (ξ i ) 1≤i≤n ; ξ i ∈ L ∞ , n  i=1 ξ i = X  (1.1) to be the set of allocations of risk X to the n traders in the market endowed with risk measures  i . Let R := {( i (X i )); (X i ) ∈ A(X )} = R(X ) (1.2) denote the corresponding risk set. Our aim is to characterize Pareto-optimal (PO) allocations (ξ i ) ∈ A(X ) i.e. allocations such that the corresponding risk vectors are minimal elements of the risk set R in the pointwise ordering. A related optimization problem is to characterize allocations (ξ i ) which minimize the total risk i.e.   i (ξ i ) = inf    i (X i ); (X i ) ∈ A(X )  (1.3) =: ∧ i (X ).