Dynamic risk-sharing game and reinsurance contract design

Abstract

This paper studies the optimal risk-sharing between an insurer and a reinsurer. The insurer purchases reinsurance for risk-control and decides her retention level with an objective to minimize her ruin probability. The reinsurer has control over the reinsurance price and aims to maximize her expected discounted profits up to the time when the insurer goes bankrupt. In a stochastic differential game-theoretic framework, we determine the insurer’s optimal reinsurance strategy and specify the reinsurance contract by solving a system of coupled Hamilton–Jacobi–Bellman equations. We obtain explicit solutions for the game problem when both the insurance and the reinsurance premiums are calculated according to the standard-deviation principle or the expected value principle, respectively. Our results show that, depending on the model parameters, the reinsurance contract is either provided with a peak price when the insurer has sufficient cash reserve and with a minimum price when otherwise, or is always provided with a peak price. We also perform some numerical analyses and provide economic interpretations for the results.