寿险模型奇异控制问题解存在唯一性及渐近解研究

寿险模型中最优投资消费及寿险购买问题是当前精算领域的重要研究课题；而带交易费用的最优投资消费问题一直是投资组合中的经典问题。项目旨在将上述两个问题结合起来，研究寿险模型中带交易费用的投资组合和寿险购买问题（在数学上，交易费用问题也称为奇异控制问题），利用并拓展随机最优控制理论研究模型中值函数的存在唯一性（即值函数是满足相应HJB方程的唯一粘性解），并进一步讨论当交易费用参数较小时，值函数和交易策略的渐近形式。同时项目将上述方法应用到死亡强度与时间相依寿险模型下，考虑生命破产概率最小化问题。项目是随机最优控制和随机分析等数学基础理论与金融保险等应用领域的交叉研究，预期研究成果对随机最优控制，鞅对偶理论等基础理论发展有重要意义，同时也为人们在现实生活中更好地进行投资消费，寿险购买提供一定理论支持和建议。

Optimal investment/consumption and life insurance purchase in life insurance models are important research issues in the actuarial science literature; Optimizing investment/consumption strategies with transaction costs is another classical problem in portfolio optimization models. The project aims to combine the above two problems to analyze life insurance models with the transaction costs and investment and life insurance purchase strategies (Mathematically, problems with transaction costs are also called singular control problems), by using and extending the stochastic optimal control theory to study the existence and uniqueness of the value function in the model (i.e., the value function is the unique viscosity solution satisfying the corresponding HJB equation), and further to discuss the asymptotic form of the value function and trading strategy when the transaction-cost parameters are sufficiently small. Moreover, in the project, we apply the above method to study life insurance models with time-dependent mortality hazard rates and consider the lifetime-ruin probability minimization problem. The project is a cross-study of fundamental mathematical theories such as stochastic optimal control and stochastic calculus, and application fields such as finance and insurance. The research results are of great significance to enhance those theories, such as stochastic optimal control and martingale duality theory and also provide theoretical support and suggestions for people to invest/consume and purchase life insurance products in reality.