Bilateral risk sharing in a comonotone market with rank-dependent utilities

This paper studies a bilateral risk-sharing problem in which the two agents are rank-dependent utility maximizers, and the market restricts risk allocations to be comonotonic. We first characterize the optimal risk allocation in an implicit way through the calculus of variations. Then, based on the element-wise maximizer of an unconstrained problem, we partition the support of loss into disjoint pieces and unveil the explicit structure of the optimal risk allocation over each piece. Our methodology reduces the dimension of the problem. We show the applicability of our results via two examples in which both agents use exponential utilities and use convex power or inverse-S-shaped probability weighting functions.