Equilibrium per-loss reinsurance strategy with delay factors and ambiguity aversion under the cooperation framework

In this paper, we investigate an optimal reinsurance problem for two ambiguity-averse insurers (AAIs) with common-shock dependence and delay factors under the utility framework. Suppose that each AAI can purchase per-loss reinsurance to reduce her claim risk, and the risk-free investment is allowed. Also, we introduce the performance-related capital inflow or outflow feature into the wealth process, which is modeled by a stochastic differential delay equation. The AAIs are supposed to be cooperative and their common objective is to find the equilibrium reinsurance strategy so as to maximize the penalized expected product of the terminal utilities. Applying techniques of stochastic control theory and corresponding Hamilton-Jacobi-Bellman-Isaacs equation, we derive the expression of the value function, and prove the existence and uniqueness of the equilibrium strategy. Numerical examples are provided to illustrate the influence of some important model parameters, which provide useful insights for reinsurance in reality. We find that the establishment of the equilibrium reinsurance strategies are affected by the claim sizes of the AAIs as well as the cooperation.