保险金融市场中相依风险模型的随机最优控制

In recent decade, the optimization problem of stochastic risk control in the modern insurance and financial market is one of the most popular research topics among the field of mathematical finance and insurance actuarial science. With the acceleration of the financial globalization progress, the risks in financial market become more and more complicated and diversity, the interdependencies among the insurance risks, the financial risks, or between the insurance risks and financial risks become stronger than before. Therefore, to construct reasonable models for all kinds of dependent risks, and to discuss the optimization problem of the stochastic risk control become the very important research topic. In this project, based on the theories of stochastic processes and stochastic control and the related methodologies, we focus on discussing the stochastic optimal control problems for the dependent risk models in the insurance and financial market. We have the innovation on the following three points: firstly, the discussion of the existence and uniqueness of the optimal values for n(≧2) classes of dependent risks in the compound Poisson risk models; secondly, the study of the stochastic optimal control problem with VaR constraints for n(≧2) classes of dependent risks not only for the compound Poisson risk models but also for the diffusion approximation risk models; the last but not the least, the investigation of the stochastic optimal control problem with bankruptcy prohibition for n(≧2) classes of dependent risks in the diffusion approximation risk models. This research not only improves the development of the theories of mathematical finance and insurance actuarial science, but also becomes the valuable reference for the decision-makers of the insurance and financial industry.

现代保险金融风险模型中的随机最优控制问题是近十年在数理金融和保险精算领域研究的前沿热点问题。随着金融全球化进程的加快，金融市场面临的风险也日益复杂和多样化，保险风险之间，金融产品之间，或者保险与金融风险之间的相互依赖关系也变得越来越强。因此，如何合理刻画保险金融市场中各类风险之间的相依关系，并探讨相依风险模型中的随机最优控制成为非常重要的研究课题。该项目至力于利用随机过程以及随机控制理论的思想来研究保险金融市场中相依风险模型的随机最优控制问题。主要在以下三个方面进行创新：一是关于n（≧2）类复合Poisson相依风险模型中最优解的存在唯一性条件的探讨；二是考虑VaR限制下n（≧2）类相依风险模型中的随机最优控制问题；三是带破产限制的n（≧2）类扩散逼近相依风险模型中的随机最优控制问题。该研究不仅可以促进数理金融和保险精算理论的发展，而且将为保险、金融行业在进行市场决策时提供很有价值的参考。

现代保险金融风险模型中的随机最优控制问题是近十年在数理金融和保险精算领域研究的前沿热点问题。随着金融全球化进程的加快，金融市场面临的风险也日益复杂和多样化，保险风险之间，金融产品之间，或者保险与金融风险之间的相互依赖关系也变得越来越强。因此，如何合理刻画保险金融市场中各类风险之间的相依关系，并探讨相依风险模型中的随机最优控制成为非常重要的研究课题。该项目至力于利用随机过程以及随机控制理论的思想来研究保险金融市场中相依风险模型的随机最优控制问题。主要在以下四个方面进行创新：一是关于n（≧2）类复合Poisson相依风险模型中最优解的存在唯一性条件的探讨；二是考虑比破产概率更一般的drawdown概率作为最优准则来探讨一系列相依风险模型中的随机最优控制问题；三是带破产限制的n（≧2）类扩散逼近相依风险模型中的随机最优控制问题；四是考虑借贷利率不一致条件下的相依风险模型中的随机最优控制问题。该研究不仅可以促进数理金融和保险精算理论的发展，而且将为保险、金融行业在进行市场决策时提供很有价值的参考。