保险公司风险管理需求下的非零和随机微分再保险博弈均衡问题研究

With the Solvency II regime came into effect, risk management becomes an important business among the insurance industry. In order to mitigate risks and enhance the solvency, insurance companies tend to enter into the reinsurance market. Therefore, the types of reinsurance contract to purchase and the amount of retained risks are significant decisions for an insurance company to make. To incorporate the competition effect among several insurance companies, researchers proceed to investigate reinsurance strategies under the framework of stochastic differential games, rather than the traditional dynamic optimization and stochastic controls for a single insurer. Though several literatures have been published, the research on stochastic differential reinsurance games is still in its early stage and there are considerable topics deserving further investigation. In this project, we study several non-zero-sum stochastic differential reinsurance games between two insurance companies under the diffusion surplus process. Depending on the practical experience of insurance companies, we investigate the combination of reinsurance and investment or capital injections strategies under appropriate objective functions. New types of risk measure will be introduced to provide facilities to measure insurers' risk exposures and hence to set up solvency constraints. Subsequently, we analyse the necessary and sufficient conditions for the existence and uniqueness of the Nash equilibrium strategies. Besides, explicit expressions for the Nash equilibrium will be derived. This project will deepen and improve the modelling for reinsurance games under risk management consideration. The Nash equilibrium obtained under various models will provide references and theoretical guidance among the decision-making process of an insurance company.

保险公司为了分散自身承担的风险和责任，同时提高自身偿付能力进而满足偿二代体系下日益重要的风险管理需求，通常选择购买再保险。如何恰当选择再保险的种类和数量是保险公司的重要决策。为反映多个保险公司间的相互竞争，再保险研究开始由传统的单个保险人的随机优化控制转向随机微分博弈框架下的纳什均衡问题。尽管国际上已经出现初步成果，但该领域的研究仍处于起步阶段，还有众多问题值得进一步探究。项目将在扩散过程下研究多个保险公司风险管理需求下的再保险、投资或者注资博弈均衡问题。根据保险公司的实务需求，提出适用于公司风险管理目标的新型风险度量，并在对应限制条件建立适合保险公司业务状况及预期目标的非零和随机微分博弈模型。进而在所建模型下分析各保险人的纳什均衡策略存在的充要条件，并给出均衡点的求解方法。课题将深化和完善风险管理下的再保险博弈模型，所建模型下得到的均衡策略将为保险公司决策提供理论依据。

本项目主要探讨保险公司风险管理需求下的再保险-投资博弈均衡和保险产品定价问题。在随机微分博弈框架下，以最大化相对盈余的期望效用为目标建立动态风险价值约束下的再保险-投资博弈模型、在模型不确定和均值方差准则下考虑模糊厌恶保险人的再保险-投资博弈、在不同保费准则下考虑具有相依风险时的均衡再保险-投资决策，进而在所建模型下分析各保险人的纳什均衡点的求解方法。在精算平衡原理下，项目研究了索赔额与索赔频率相依时的车险、政府兜底的 PBGC 两类产品定价的精算实务问题。项目深化和完善了风险管理下的再保险博弈模型，并为理论与实践的结合搭建桥梁，所建模型下得到的均衡策略有望为保险公司决策提供理论依据。此外，项目提出的动态风险价值对于偿二代下的保险公司准备金计提具有一定的科学意义。