This paper considers a non-zero-sum stochastic differential game between two competitive mean-variance insurers, who aim to seek the time-consistent reinsurance and investment strategies. These two insurers are allowed to purchase proportional reinsurance to mitigate individual claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the constant elasticity of variance model. The main objective of each insurer is to maximizing the mean-variance utility of his relative terminal wealth with respect to that of his competitor. Applying the techniques of stochastic control theory, we derive the Nash equilibrium reinsurance and investment strategies explicitly and present the corresponding verification theorem. Furthermore, Nash equilibrium strategies and value functions are also provided under the diffusion approximation model. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.

本文考虑了两家竞争均值方差保险公司之间的非零和随机微分博弈，旨在寻求时间一致的再保险和投资策略。允许这两家保险公司按比例购买再保险，以降低个人索赔风险；并且可以投资一种无风险资产和一种价格动态遵循恒定方差弹性模型的风险资产。每个保险公司的主要目标是最大化其相对最终财富相对于其竞争对手的平均方差效用。应用随机控制理论的技术，我们明确地推导出了纳什均衡再保险和投资策略，并给出了相应的验证定理。此外，在扩散近似模型下还提供了纳什均衡策略和价值函数。最后，我们通过一些数值例子来说明模型参数对均衡再保险和投资策略的影响，并从这些结果中得出一些经济解释。