正倒向随机微分方程高阶方法的构造和分析

正倒向随机微分方程在金融数学、随机最优控制以及风险度量等前沿领域都有重要应用。目前，对于正倒向随机微分方程，有严格理论保障的高阶高效数值方法还很少。. 本项目主要研究正倒向随机微分方程高阶方法的构造、分析以及快速实现。首先，深入研究正倒向随机微分方程以及相关的常微分参考方程，构造新的计算成本较低的高阶数值方法。其次，针对正倒向随机微分方程高阶数值方法理论分析异常困难的问题，引进或提出新的理论分析工具和技术，为新构造的数值方法建立相关理论证明框架。最后，结合新构造的数值方法的特性，设计恰当的条件数学期望逼近算法，进而提高新方法的实际计算效率。. 本项目旨在对正倒向随机微分方程高阶高效数值方法进行新的探索，为一些实际问题提供有理论保障的高效数值方法。

Forward backward stochastic differential equations have important applications in frontier fields, such as financial mathematics, stochastic optimal control and risk measurement. At present, there are few high-order and efficient numerical methods with rigorous theoretical guarantee for the forward backward stochastic differential equations.. This project focuses on the construction, analysis and fast implementation of high-order methods for the forward backward stochastic differential equations. Firstly, forward backward stochastic differential equations and related ordinary differential reference equations will be deeply studied to construct some new high-order numerical methods with low computational cost. Secondly, in view of the extremely difficult problems in the theoretical analysis of high-order numerical methods for the forward backward stochastic differential equations, new theoretical analysis tools and techniques will be introduced or proposed to establish a framework of relevant theoretical proofs for the newly constructed numerical methods. Finally, combined with the characteristics of the new methods, the appropriate conditional expectation approximation algorithms will be designed to improve the actual calculation efficiency of the new methods.. This project aims to make a new exploration for the high-order and efficient numerical methods of forward backward stochastic differential equations, and to provide theoretically guaranteed efficient numerical methods for some practical problems.