Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks

This paper investigates a non-zero-sum stochastic differential game between two mean–variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The non-zero-sum goal of these insurers is to maximize the mean–variance utility of a weighted value of their terminal and relative wealths. We solve the mean–variance problem in the time-consistent case and establish the extended Hamilton−Jacobi−Bellman systems for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies. •Study a time-consistent mean–variance reinsurance and investment game problem.•Consider the insurance risk, volatility risk and default risk.•Provide verification theorem at pre-default and post-default cases.•Derive explicit solutions of equilibrium strategies.•Present detailed sensitivity analysis to show the insurers’ economic behaviors.