带跳弱耦合正倒向随机微分方程的Crank-Nicolson类型数值解法研究

In this project, we will study the most direct Crank-Nicolson-type numerical methods for solving the weakly coupled forward backward stochastic differential equations (short for FBSDEs) with jumps: from Malliavin stochastic analysis for Brownian motion, combining the numerical methods for solving the forward stochastic differential equations, using the relationship between FBSDEs and PDEs, we will obtain the stability convergence and error estimates of the Crank-Nicolson scheme for solving the weakly coupled FBSDEs and weakly coupled FBSDEs with jumps. Then we will propose some new kinds of Crank-Nicolson scheme for solving the weakly coupled FBSDEs with jumps, then using the Malliavin stochastic analysis for Lévy processes and numerical methods for solving forward stochastic differential equations, we will rigorously analyze their stability, convergence and error estimates for solving the weakly coupled FBSDEs with jumps; Finally, we will apply the proposed methods to the simulation and computation of some important financial models.

本项目拟研究带跳弱耦合正倒向随机微分方程(简称FBSDEs)的Crank-Nicolson类型数值解法: 通过布朗运动的Malliavin随机分析，结合求解正向随机微分方程数值方法，利用FBSDEs解和偏微分方程解的关系，得到求解弱耦合FBSDEs和带跳弱耦合FBSDEs的最直接Crank-Nicolson格式的稳定性、收敛性和理论误差估计；提出一些新的求解带跳弱耦合FBSDEs的Crank-Nicolson类型格式，并通过Lévy过程的Malliavin随机分析和带跳正向随机微分方程数值方法，对其稳定性、收敛性和理论误差估计进行严格的分析；应用所提方法于一些重要的金融模型的模拟和计算。

本项目研究了带跳正倒向随机微分方程 (简称FBSDEs) 的Crank-Nicolson类型数值解法: 通过布朗运动的Malliavin随机分析，结合求解正向随机微分方程数值方法，利用FBSDEs解和偏微分方程解的关系，得到求解 FBSDEs的最直接Crank-Nicolson格式的稳定性、收敛性和理论误差估计；提出了一些新的求解带跳FBSDEs的Crank-Nicolson类型格式，并通过Lévy过程的Malliavin随机分析和带跳正向随机微分方程数值方法，对其稳定性、收敛性和理论误差估计进行严格的分析；应用所提方法于一些重要的金融模型的模拟和计算。