This paper studies a non-zero-sum stochastic differential game for multiple mean-variance insurers. Insurers can purchase proportional reinsurance and invest in a risk-free asset, a market index, a defaultable bond and multiple pairs of mispriced stocks. The dynamics of the mispriced stocks satisfy a “cointegrated system” where the expected returns follow the mean reverting processes, and the bond is defaultable with a recovering proportional value at default. In particular, we assume that the investment opportunities in mispriced stocks are only available for a few insurers, which is more realistic and in line with the superiority of information in the competitive market. Each insurer's objective is maximizing a function of her terminal wealth and competitors' relative wealth under the mean-variance criterion. Using techniques in stochastic control theory, we establish the extended Hamilton-Jacobi-Bellman equations and obtain the equilibrium strategies. Note that the derived solutions are analytical and time-consistent, and we verify the competitive advantages gained from investment opportunities in mispriced stocks. We represent our results in terms of the M-matrices, which help us prove the existence and uniqueness of the solutions and further explicitly analyze how the crucial arguments in the model affect the equilibrium strategies. Numerical examples with detailed sensitivity analyses are presented to support our conclusions.

本文研究了多个均值方差保险公司的非零和随机微分博弈。保险公司可以购买比例再保险并投资于无风险资产、市场指数、可违约债券和多对错误定价的股票。错误定价股票的动态满足“协整系统”，其中预期收益遵循均值回归过程，并且债券可违约，违约时价值恢复成比例。特别是，我们假设错误定价股票的投资机会只为少数保险公司提供，这更加现实，也符合竞争市场中的信息优势。每个保险公司的目标是在均值方差标准下最大化其终端财富和竞争对手相对财富的函数。利用随机控制理论中的技术，我们建立了扩展的 Hamilton-Jacobi-Bellman 方程并获得了均衡策略。请注意，导出的解决方案是分析性的且时间一致的，并且我们验证了从错误定价股票的投资机会中获得的竞争优势。我们用 M 矩阵表示我们的结果，这有助于我们证明解的存在性和唯一性，并进一步明确分析模型中的关键参数如何影响均衡策略。给出了带有详细敏感性分析的数值例子来支持我们的结论。