跳驱动的正倒向随机微分方程的数值方法及其应用研究

Solving the jump driven forward backward stochastic differential equations (FBSDEJ) is an urgent need for their applications. The random complexity of FBSDEJ makes it difficult to obtain its solution, and its numerical solution is an important part of the theory of forward backward stochastic differential equations (FBSDE) and is the frontier in the field of international scientific computing. Based on the theories of FBSDE, partial differential equations, stochastic analysis and scientific computing, and on the previous works on numerical solutions of FBSDE, this proposal will do researches on numerical methods for FBSDEJ and their applications. Propose highly effective and accurate new one-step and multi-step numerical methods; rigorously analyze the efficiency, stability, convergence and errors of the proposed methods theoretically and numerically; study their applications in solving fractional differential equations, nonlocal diffusion and financial problems. The project is of very theoretical significance and application value to further understand the mechanism of FBSDEJ, improve the scientific computing theory of FBSDE, and accelerate the application of FBSDE theory in finance, nonlocal diffusion problems and related fields.

跳驱动的正倒向随机微分方程(FBSDEJ)的求解是其在相关应用研究领域发展的迫切需求。FBSDEJ本身结构的复杂性使得求解析解十分困难，其解的科学计算是正倒向随机微分方程(FBSDE)理论的重要组成部分，是国际科学计算的前沿课题。本项目拟基于正倒向随机微分方程、偏微分方程、随机分析和科学计算等理论，以及前期关于FBSDE数值解法的研究，展开跳驱动的正倒向随机微分方程的数值方法及其应用研究：提出求解FBSDEJ新的高效高精度单步数值方法和多步数值方法；严格理论数值分析所提方法的有效性、稳定性和收敛性，得到数值方法的误差估计；研究所提数值方法在分数阶微分方程、非局部扩散及金融等问题的应用。本项目的研究对深入理解FBSDEJ的机理，完善FBSDE的科学计算理论，加快拓宽正倒向随机微分方程理论在金融、非局部扩散问题及相关领域中的应用，都有十分重要的理论意义和应用价值。

正倒向随机微分方程在随机最优控制、金融数学、偏微分方程理论、非线性期望等众多研究领域有重要应用。正倒向随机微分方程科学计算方法的研究是计算数学领域的热门研究方向。本项目主要研究了求解正倒向随机微分方程的概率数值格式和统一的数值理论分析框架；提出求解二阶正倒向随机微分方程的显式递延校正数值方法；给出一种新的求解非耦合正倒向随机微分方程的二阶数值格式并证得最优误差估计；研究了求解平均场情形下正倒向随机微分方程的显式多步数值方法。研究成果发表在《SIAM Journal on Numerical Analysis》《Journal of Scientific Computing》《Journal of Computational Mathematics》《East Asian Journal on Applied Mathematics》等计算数学领域的著名期刊。