Robust reinsurance contract with asymmetric information in a stochastic Stackelberg differential game

In this paper, we determine a robust reinsurance contract from joint interests of the insurer and reinsurer under the framework of Stackelberg differential game. More specifically, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. In order to defend the large shocks of wealth process, a loss-dependent premium principle is applied to the insurer. Meanwhile, we incorporate model uncertainty into the reinsurer's controlled surplus due to the asymmetric information. Under the time-consistent mean-variance criterion, we derive the robust reinsurance contract explicitly by solving the coupled extended Hamilton–Jacobi–Bellman systems. It is interesting to prove that the optimal premium control for the reinsurer is determined by a time-adjusted variance principle. In addition, we find that the reinsurer would like to raise the reinsurance price to guard against the model uncertainty, which consequently decreases the insurer's reinsurance demand. Finally, further analyses are provided to show the necessity of considering the model uncertainty; otherwise, the reinsurance company will suffer a great loss of utility.