n-Agent reinsurance and investment games for mean-variance insurers under multivariate 4/2 stochastic covariance model

In this study, we investigate the time-consistent equilibrium reinsurance-investment strategy for n competitive insurers under the mean-variance criterion. Each insurer can purchase proportional reinsurance for reducing the claim risk and invest in the financial market for increasing his wealth. The surplus process of each insurer is characterized by the approximate diffusion process, and considering the stochastic correlation between risky assets, we introduce a multivariate 4/2 stochastic covariance model to characterize the price processes of two risky assets. The objective of each insurer is to find the optimal investment-reinsurance strategy so as to maximize the expected value of its terminal relative wealth while minimizing the variance of the terminal relative wealth. By introducing an auxiliary deterministic process, we obtain a modified mean-variance objective function and construct an alternative time-consistent mean-variance control problem related to the original mean-variance problem. Subsequently, by solving the Hamilton-Jacobi-Bellman (HJB) equation, the time-consistent equilibrium investment-reinsurance strategies for insurers are derived. Numerical examples are presented at the end of the article to demonstrate the effects of different parameters on the equilibrium strategies. The results reveal that competition between insurers will encourage them to adopt more aggressive investment and reinsurance strategies; and meanwhile, the correlation between risky assets in financial markets will also influence the decision-making of insurers.