Stackelberg-equilibrium reinsurance contract with heterogeneous beliefs and uncertain parameter

This paper aims to explore the robust reinsurance contract design with belief heterogeneity in the framework of the Stackelberg differential game. In this setup, the reinsurer plays the role of leader to decide the optimal reinsurance premium, while the insurer acts as the follower to determine the optimal retention level under proportional reinsurance arrangements. It is assumed that the insurer and the reinsurer have heterogeneous beliefs and adopt distinct distributions for the aggregate risk due to asymmetric information and different estimation methods. Moreover, we suppose that the reinsurer applies interval estimation to approximate the expected value of aggregate claims, denoted as $ \mu_2 $, within an interval $ [\hat{\mu}_2-\delta,\hat{\mu}_2+\delta] $, where $ \hat{\mu}_2 $ represents the estimated value and $ \delta $ signifies the estimation error. Under the time-consistent mean-variance criterion, we derive reinsurance contracts for both ambiguity-averse and ambiguity-loving reinsurers by solving the corresponding extended Hamilton-Jacobi-Bellman systems. Furthermore, to provide a deeper understanding of the influence of heterogeneous beliefs on the optimal reinsurance contract and value functions, we focus on $ \delta = 0 $ and derive several intuitive results.