In this paper, we investigate a non-zero-sum stochastic differential reinsurance-investment game problem between two insurers. Both insurers can purchase proportional reinsurance and invest in a financial market that contains a risk-free asset and a risky asset. We consider the insurers' wealth processes with delay to characterize the bounded memory feature. For considering the effect of asymmetric information, we assume the insurers have access to different levels of information in the financial market. Each insurer's objective is to maximize the expected utility of its performance relative to its competitor. We derive the Hamilton-Jacobi-Bellman (HJB) equations and the general Nash equilibrium strategies associated with the control problem by applying the dynamic programming principle. For constant absolute risk aversion (CARA) insurers, the explicit Nash equilibrium strategies and the value functions are obtained. Finally, we present some numerical studies to draw economic interpretations and find the following interesting results: (1) the insurer with less information completely ignores its own risk aversion factor, but imitates the investment strategy of its competitor who has more information on the financial market, which is a manifestation of the herd effect in economics; (2) the difference between the effects of different delay weights on the strategies is related to the length of the delay time in the framework of the non-zero-sum stochastic differential game, which illustrates that insurers should rationally estimate the correlation between historical performance and future performance based on their own risk tolerance, especially when decision makers consider historical performance over a long period of time.

中文翻译：

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具有时滞和信息不对称的非零和再保险投资博弈

在本文中，我们研究了两家保险公司之间的非零和随机差异再保险投资博弈问题。两家保险公司都可以购买比例再保险，并在一个包含无风险资产和风险资产的金融市场中进行投资。我们认为保险公司的财富过程会延迟以刻画有限记忆特征。考虑到信息不对称的影响，我们假设保险公司可以在金融市场上访问不同级别的信息。每个保险公司的目标是相对于其竞争对手，最大化其业绩的预期效用。通过应用动态规划原理，我们推导了汉密尔顿-雅各比-贝尔曼（HJB）方程和与控制问题相关的一般纳什均衡策略。对于持续的绝对风险规避（CARA）保险公司，得到了明确的纳什均衡策略和价值函数。最后，我们进行一些数值研究以得出经济解释，并得出以下有趣的结果：（1）信息较少的保险公司完全忽略了其自身的风险规避因素，而模仿了在金融市场上拥有更多信息的竞争对手的投资策略。 ，这是经济学中羊群效应的体现；（2）在非零和随机微分博弈框架下，不同延迟权重对策略的影响之间的差异与延迟时间的长短有关，这说明保险公司应合理估计历史绩效与历史绩效之间的相关性。以及基于自身风险承受能力的未来表现，