模型不确定下几类跳跃风险模型的随机控制问题与数值计算

Stochastic control problem has been a hotspot in actuarial research．It is an innovative and challenging scientific problem that the robust optimal control problem considering the impact of model uncertainty on the robustness of the optimal strategies．This project aims at three different types of jump models，by using techniques of stochastic control theory, we study the robust optimal control problems under the framework of model uncertainty，and explore the control strategies which are acceptable and more robustness for the insurance companies．The research mainly includes three parts：1．Assuming that the insurance risk models with jumps，and considering the investment and reinsurance strategies，we focus on investigate the robust optimal control problems and numerical calculation methods when the analytical solutions can not be obtained；2．We study the robust optimal investment and reinsurance problems in the financial markets models with jumps，especially to calculate the robust optimal investment and reinsurance strategies with default risk under variety of objectives；3．We study the robust optimal control problems under Markov-modulated financial markets models and insurance risk models，and discuss the corresponding numerical solutions．This research combines well with the robust optimization theory and the stochastic control theory to solve the focus issue of stochastic control problem in the field of actuarial，which has important theoretical value and practical significance．

随机控制问题一直是保险精算领域的研究热点，考虑模型不确定性对最优策略稳健性的影响的鲁棒最优控制问题是其中兼具创新性和挑战性的科学问题。本项目围绕三种不同类型的跳跃模型，以随机控制理论为技术手段，研究模型不确定框架下的鲁棒最优控制问题，寻找保险公司可以承受的，同时又更具有稳健性的控制策略。研究主要包括三个部分：1．在带跳跃项的保险风险模型下，考虑投资和再保险策略，重点研究不能获得解析解时的鲁棒最优控制问题的数值计算方法；2．在带跳跃项的金融市场模型下，研究保险公司的鲁棒最优投资-再保险问题，侧重在不同优化目标下计算带违约风险的鲁棒最优投资-再保险策略；3. 研究保险公司在马氏调制的金融保险模型中的鲁棒最优控制问题及其数值求解。本研究将鲁棒优化理论与随机控制理论很好地融合并用以解决保险精算领域中热门的随机控制问题，具有重要的理论价值和现实意义。

随机控制问题一直是保险精算领域的研究热点，考虑模型不确定性对最优策略稳健性的影响的鲁棒最优控制问题是其中兼具创新性和挑战性的科学问题。本项目主要绕“终端时刻财富期望效用最大化”和“均值-方差准则”这两个优化目标，把“相对熵”惩罚的鲁棒控制思想与随机控制理论结合起来，在不同的金融保险模型中，研究了保险公司和再保险公司的稳健最优投资、再保险及风险控制策略，得到了相应的解析结果，具有重要的理论价值和现实意义。