Optimal Stochastic Derivative Proportional Reinsurance and Investment Game Problem With Delay in a Defaultable Market

In this paper, we investigate a non–zero‐sum stochastic differential investment and reinsurance game problem between two competitive insurers with delay and default risk. The two insurers have access to a risk‐free asset: a risky asset and a defaultable bond. The reinsurance premium is calculated based on the expected premium principle, and the goal of each insurer is to maximize the expected value of the relative performance between insurers and minimize the variance of the relative wealth at the end moment. Under the mean–variance criterion, we also use a game theory framework to establish the optimal investment–reinsurance problem. By applying stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations, the explicit expressions of the robust equilibrium investment–reinsurance strategy and the corresponding equilibrium value function are obtained. Finally, numerical examples are given to elaborate the optimal investment–reinsurance strategy, and the effects of model parameters are discussed.