马尔科夫调节风险模型下有关保险精算的几个随机微分博弈问题

This project aims to address two kinds of problems related to stochastic control and stochastic differential games under Markov modulated risk model. One topic is to study optimization problems when there are two players, zero-sum games. The other one is to study optimal control problems for partially observed (also known as incomplete information) Markov modulated model. Stochastic control problems for Markov modulated process is of great interest in stochastic optimization and operation research, thus the topics to be studied in this project have attracted a vast of attentions recently due to their significance in mathematical theoretical study. On the other hand, these topics are also highly concerned in risk management and risk control. Thus，this project is of both practical and theoretical value. The main idea we adopted is to explore and apply the theory of stochastic differential games for solving the problems proposed in this proposal. The content to be studied in this project includes two main aspects..(1)Finish the proof for the existence of Nash equilibrium points of stochastic differential games under underlined optimization criterion; when there exists constrains on the game strategies, finish the proof for the existence of value function; find the analytical solution of optimal strategies and value function; when it is not able to get the analytical value function, try to construct weak convergence series for approximating the optimal strategies and value function .(2)Address optimization problems for partially observed Markov modulated risk model, include: optimal investment, optimal reinsurance and optimal dividend. The main work here is to explore and apply stochastic differential games for solving these problems. Our idea is to get a kind of robust optimal strategy by finding the equilibrium solution between the decision maker and the incomplete information and to derive robust optimal strategy..Stochastic control for Markov modulated risk model has been widely studied recently, but there are still many meaningful problems to be studied. We believe that the implementation of this project will promote the development and applications of stochastic differential games for Markov modulated risk model.

马尔科夫调节风险模型的最优化问题在保险精算学尤其是风险理论中有着广泛应用。其中，完全观测的该模型下存在博弈时的最优化问题以及不完全观测的该模型下若干最优化问题的随机微分博弈方法是两个很有意义的课题。本项目拟对这两个问题展开研究，主要研究内容包括：拟发展HJBI方程的有关结果，结合粘性解理论给出马尔科夫调节风险模型的有关微分博弈问题值函数的存在性并完成验证性证明；讨论最优解的解析性，当没有解析解时，结合随机过程的弱收敛理论，完成最优解的算法设计与理论分析；对若干受约束的随机博弈问题，证明其值函数的存在性并展开相关数值方法的理论分析；对不完全观测的马尔科夫调节风险模型，通过信息与决策者之间的微分博弈，得到稳健的最优控制并完成相关理论分析。本项目不仅可以推动马尔科夫调节模型下随机微分博弈理论的基础研究，还可以进一步拓宽该理论在风险理论中的应用范围，提升其应用层次，因而兼具理论学术意义和应用价值。

本项目主要研究保险精算中马尔科夫调节模型下几个随机控制问题与随机微分博弈问题。具体包括：马尔科夫调节模型的最优投资与微分博弈问题，给出了博弈问题的值函数满足的HJBI方程并给出均衡点存在性的证明；针对前述博弈问题，给出了一般情形下求解值函数和最优解的数值方法；研究了马尔科夫调节风险模型的分红注资问题，分红问题涉及到分红最大化与破产概率约束之间的博弈，给出了分红注资问题最优解的显式表达式；研究了连续时间马尔科夫调节风险模型的最优投资以及再保险问题，在理赔服从指数分时，给出了最优解的显式表达式；研究了马尔科夫调节模型下有高水位税费支付时的投资消费问题，此时一般难以得到最优解的显式表达式，我们得到了值函数和相关HJB方程粘性解的关系，当模型受到马尔科夫过程调节的时候，证明了随着伴随马尔科夫过程状态变化的投资和消费策略是最优的；通过最小化破产概率上界的方法给出了具有马尔科夫调节的离散化模型的最优投资分配问题，此外我们还研究了一般分数布朗运动刻画市场下权益指数年金的定价问题。项目的主要内容已经研究完成，项目课题组成员共发表科研论文18篇，其中外文期刊论文10篇，外文期刊中，被SCI检索论文8篇，被SSCI检索论文6篇，国内核心期刊论文8篇，投稿在审论文2篇。论文被SCI期刊他引11次。项目组成员积极参加国内外学术交流活动，获得安徽省科学技术奖三等奖一项。利用本项目的基金资助与驱动，课题组成员还提升了研究生的人才培养质量，共计培养毕业5名研究生，在读研究生7名。