Pareto Optimal Allocations and Optimal Risk Sharing for Quasiconvex Risk Measures

Pareto optimal allocations and optimal risk sharing for coherent or convex risk measures as well as for insurance prices have been studied widely in the literature. In particular, Pareto optimal allocations have been characterized by applying inf-convolution of risk measures and convex analysis.In the recent literature, an increasing interest has been devoted to quasiconvex risk measures, that is risk measures where convexity is replaced by quasiconvexity and cash-additivity is dropped.The main goal of this paper is then to generalize the characterization of Pareto optimal allocations known for convex risk measures (see, among others, Jouini et al.) to the quasiconvex case. Following the approach of Jouini et al. for convex risk measures, in the quasiconvex case we provide su cient conditions for allocations to be (weakly) Pareto optimal in terms of exactness of the so-called quasiconvex inf-convolution. Moreover, we give a necessary condition for weakly optimal risk sharing that is also su cient under cash-additivity of at least one between the risk measures