Equilibrium reinsurance strategies for catastrophe and secondary claims under α-maxmin mean–variance criterion

This paper investigates optimal reinsurance under the consideration of contagious catastrophe claims and secondary claims, and the intensity of the latter is modeled as a shot noise process impacted by the former. Also, an α-maxmin mean–variance (MV) criterion is adopted to allow the insurer to have different levels of ambiguity aversion attitudes, and the general mean–variance premium principle is applied to calculate the reinsurance premiums. To overcome the time-inconsistency issue in the problem, this paper solves the optimization problem via studying the corresponding Extended Hamilton–Jacobi–Bellman (EHJB) equation. With the help of some auxiliary problems, the existence and uniqueness of the optimal reinsurance strategies are demonstrated. Our research indicates that an insurer’s reinsurance strategy for catastrophe claims is influenced by the strategy for secondary claims, but not vice versa. Additionally, it is observed that, for catastrophe insurance businesses, the proportional reinsurance contract is optimal when applying variance premium principle, while the excess-of-loss reinsurance treaty is generally preferred under the expected value premium principle. Finally, the sensitivity analysis for optimal reinsurance strategies with respect to several model parameters are performed.