Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide a complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L ^p for any p∈[1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz-continuous functions of the aggregate risk. Filipović is supported by WWTF (Vienna Science and Technology Fund). Svindland gratefully acknowledges financial support from Munich Re Grant for doctoral students and hospitality of the Research Unit of Financial and Actuarial Mathematics, Vienna University of Technology. We thank Beatrice Acciaio and Walter Schachermayer for fruitful discussions and an anonymous referee for helpful remarks.