具有时间延迟的分红策略与最优控制

The project focuses on the dividend strategies with time delay and the optimal control problem for the MAP risk process. There are two kinds of delay in dividends, one is the dividend can only be paid on some observation times, the other is the Parisian delay. The content in this project includes two parts: (1) Both the threshold strategy and barrier strategy are studied in the MAP risk model with more frequent ruin monitoring than dividend decisions. The affine dividend strategy is discussed when the observation times are exponentially distributed. And the optimal dividend strategies are also discussed in some special MAP risk models. (2) The barrier strategy, affine strategy and impulse strategy with Parisian implementation delays for the MAP risk model are discussed. Moreover, capital injections, investment, debit interest, transaction costs and other financial factors will be applied. To discuss the optimal dividend strategies, stochastic process, dynamic programming, optimal control, numerical analysis and differential game theory will be used. The research content is the frontier in actuarial field, which has important values in theory and application.

本项目主要是在MAP(马氏到达过程)风险模型中对有时间延迟的分红策略与最优控制问题的研究。分红中的延迟行为有两种，一种是分红只能发生在某些随机观察时刻上的延迟，另一种是Parisian延迟。本项目的主要研究内容包括两个方面：一是在对破产的监控比对分红的监控更频繁的情况下研究MAP模型的阈限分红和障碍分红；在指数观察时间时研究仿射分红及某些特殊MAP模型的最优分红策略。二是有Parisian延迟的障碍分红、仿射分红和脉冲分红的研究。同时，注资、贷款、投资、交易费等金融因素也会被考虑进来。本项目将利用随机过程、动态规划原理、最优控制理论、数值分析、微分博弈论等技术方法研究模型的最优分红策略。课题所研究的内容是当前国际精算界的前沿问题，具有很高的理论价值和应用价值。

本项目主要是研究了一些MAP风险模型中有时间延迟的分红问题与最优控制问题。分红中的延迟既包括随机延迟也包括固定时间的Parisian延迟。本项目所做的主要工作有三个方面：(1) 随机观测时间下风险模型的研究，包括Levy风险模型的周期分红、注资、分红次数的联合分布；有借贷的经典模型的最优周期分红；均衡分红策略等。(2)在有Parisian固定时间延迟的情况下研究了MAP风险模型中的Parisian破产概率以及有投资借贷的经典模型的Parisian延迟分红问题。(3)在随机观测与Parisian延迟相结合的混合观测体制模型中研究了Parisian延迟破产与分红问题。本项目所研究的模型与问题比较贴合实际生活，是保险精算的热点，具有很高的理论价值和应用价值。