保险风险控制理论以及养老金问题的研究

In this project, we study the dividend, reinsurance and investment problems for some insurance risk models and pension funding problems. It mainly includes the following specific topics: Ruin probability, Gerber-Shiu function, optimal reinsurance, optimal dividends and optimal investment strategies for the insurance risk models driven by G-Brownian motion; When the Poisson shot noise process is incorporated in the pricing process of the risky asset, it means that the jump effect of the price will fade away which is more practical than the classical geometric Brownian motion price process. The optimal dividend, reinsurance and investment strategies for this type of insurance risk process will be studied in this project; The third problem is about the study of the statistical properties of some actuarial qualtities such as ruin, Gerber-Shiu function, expected discounted dividend, etc. The consistency, optimality and asymptotic properties will be the main focus of this part. Finally, life insurance risk models, pension fund problems will be studied by using the stochastic control theory.

本项目研究保险风险理论中的投资分红及再保险问题以及养老基金问题。主要包括G-布朗运动驱动的风险模型的破产概率，Gerber-Shiu函数，最优分红，最优投资及再保险；当风险资产的价格过程由泊松Shot noise过程驱动时，相应的风险过程在不同目标和限制条件下的投资、分红以及再保险策略；结合随机控制理论研究寿险中的投资、消费和养老金问题；精算量的统计分析，包括经典精算量如破产概率，Gerber-Shiu函数等，以及近年来课题组在保险中的随机最优控制领域所取得的系列成果中的重要精算量如期望折现分红等的估计并研究估计的相合性，最优性及收敛速度等问题。

本项目主要开展保险风险理论中的随机最优控制问题研究，其中包括最优投资和最优再保险问题，更新风险模型的最优分红问题，精算量的估计问题，DB和DC型养老金问题，与G-期望相关的最优投资等问题。通过项目组成员的分工合作，取到了一批高质量的研究成功。其中包括了相依风险模型的最优投资和再保险问题，具有随机波动率情形下的期望和方差问题，几种新的极大值原理的建立，最优分红问题，并给出了破产概率的新的估计方法以及与养老金相关的最优投资策略。项目取得的成果不仅在理论上有重要的突破，而且具有潜在的实际应用价值。项目组成员经过努力，圆满完成了项目的任务。