本文基于均值-方差准则, 研究了一个保险公司与一个再保险公司之间竞争下的鲁棒最优再保险问题。保险公司经营$ n $种相依保险业务, 它对每种保险业务购买再保险来减少索赔风险。通过相对业绩, 本文量化了保险公司与再保险公司之间的竞争。保险公司的目标是, 在最坏市场情形下, 给定终端财富的均值时, 选择最优再保险策略使其面临的风险最小。通过应用随机控制和随机动态规划理论, 建立了Hamilton-Jacob-Bellman-Isaacs (HJBI)方程。进而, 通过求解HJBI方程, 并利用拉格朗日对偶理论, 本文得到了鲁棒最优再保险策略的解析解。最终, 通过数值实验解释了模型参数对鲁棒最优再保险策略和有效前沿的影响。研究结果可以指导保险公司在经营多种保险业务时, 采取最优再保险策略, 使其面临的风险最小。

Based on the mean-variance criterion, this paper studies the robust optimal reinsurance problem under the competition between an insurance company and a reinsurance company. The insurance company operates $ n $ kinds of dependent insurance businesses, and it buys reinsurance for each insurance business to reduce the claim risk. Through relative performance, this paper quantifies the competition between the insurance company and the reinsurance company. The insurance company's goal is to choose an optimal reinsurance strategy to minimize the risk when the mean of terminal wealth is given in the worst market situation. By using the theory of stochastic control and stochastic dynamic programming, this paper establishes the Hamilton-Jacob-Bellman-Isaacs (HJBI) equation. Furthermore, by solving HJBI equation and using Lagrange duality theory, this paper obtains the explicit solution for the robust optimal reinsurance strategy. Finally, the influence of model parameters on the robust optimal reinsurance strategy and efficient frontier is explained by numerical experiments. The research results can guide insurance companies to adopt optimal reinsurance strategies to minimize the risks that they face when operating a variety of insurance businesses.