Pareto optimal allocations and optimal risk sharing for quasiconvex risk measures

The main goal of this paper is to generalize the characterization of Pareto optimal allocations known for convex risk measures (see, among others, Jouini et al., in Math Financ 18(2):269–292, 2008 and Filipovic and Kupper, in Int J Theor Appl Financ, 11:325–343, 2008) to the wider class of quasiconvex risk measures. Following the approach of Jouini et al., in Math Financ 18(2):269–292, 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 as well as an existence result for weakly Pareto optimal allocations. 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.