Abstracts

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, where convexity is replaced by quasiconvexity and cash-additivity is dropped. The main motivation to the introduction of such risk measures is that the right formulation of diversification of risk is quasiconvexity (instead of convexity) (see Cerreia-Vioglio et al. (2011), Drapeau and Kupper (2010) and Frittelli and Maggis (2011)).
The main goal of this paper is then to generalize the characterization of Pareto optimal allocations and of optimal risk sharing known for convex risk measures (see, among others, Jouini, Schachermayer and Touzi (2008)) to the quasiconvex case. Following the approach of Jouini et al. (2008) for convex risk measures, in the quasiconvex case we provide sufficient conditions for allocations to be (weakly) Pareto optimal in terms of exactness of the so-called quasiconvex inf-convolution.
We provide also some counterexamples showing that exactness of the qco-convolution does not guarantee that any allocation attaining the infimum in the qco-convolution is Pareto optimal, but only weakly Pareto optimal; and that weakly Pareto optimality does not imply in general exactness.
Moreover, we give a necessary condition for weakly optimal risk sharing that is also sufficient under cash-additivity of (at least) one between the risk measures.