本文研究了VaR约束下两个竞争保险公司的最优再保险-投资策略。我们假设保险公司的动态盈余过程用经典的Cramer-Lundberg (C-L)风险模型来描述，该模型中的保费由损失相依保费原则确定。此外，保险公司可以购买比例再保险并投资于一个由一个无风险资产和一个风险资产组成的金融市场，其中风险资产的价格过程由几何布朗运动描述。首先，我们以保险公司相对终端财富的期望效用最大化为目标，建立了VaR约束下的优化问题。接下来，我们利用最优控制理论和动态规划原理解决了相应的约束优化问题。特别地，我们在指数效用下得到了三种不同情形下的纳什均衡策略。最后，通过具体的数值分析，阐述了一些参数对最优再保险策略和最优投资策略的影响，并得到了一些有意义的结论。

This paper investigates the optimal reinsurance-investment strategies of two competing insurers under VaR constraints. We assume that the insurers' dynamic surplus processes are described by the classic Cramer-Lundberg (C-L) risk model, in which the premiums are determined by the loss-dependent premium principle. Moreover, the insurers can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset, where the price process of the risky asset is described by the geometric Brownian motion. Firstly, we aim to maximize the expected utility of the insurers' relative terminal wealth and then establish optimization problems with the VaR constraints. In the next, we solve the corresponding constrained optimization problems by using the optimal control theory and the dynamic programming principle. Specially, we get three different Nash equilibrium strategies under exponential utility. Finally, we illustrate the effects of some parameters on the optimal reinsurance strategy and the optimal investment strategy through specific numerical analysis, and find some interesting results.