G-Brown运动是G-期望意义下的一类均值为零，方差不确定的随机过程，具有丰富有趣的新结构不是经典Brown运动的一般推广，其研究具有重要的科学意义和实际应用前景。目前，G-Brown运动相关问题的研究还处于初级阶段，理论尚不完善，特别是数值收敛性、数值稳定性等研究比较少。带有Poisson跳过程的随机微分方程的研究相对成熟，但是关于建立其真实解的稳定性和其数值解的稳定性之间的充分必要条件的研究却很少。本项目首先讨论G-Brown运动的数值模拟，进而用数值方法模拟G-Brown运动驱动的随机微分方程，从而对其数值分析，研究其收敛性和稳定性等。进一步，本项目建立G-Brown运动驱动的随机微分方程及带有Poisson跳过程的随机微分方程真实解的稳定性和数值解的稳定性之间的充分必要条件，从而在一定情形下避免了构造Lyapunov函数研究随机微分方程稳定性带来的困难。期望本项目研究有新突破。

The G-Brownian motion is a zero-mean and variance-uncertainty stochastic process in the sense of G-expectation. It has a rich and interesting new structure which non-trivially generalizes the classical Brownian motion. So, it has great applications and significance in science. At present, the G-Brownian motion is still in its infancy, and the related theories are not complete, especially the numerical convergence and numerical stability have been little studied. As we know, the research of stochastic differential equations with Poisson jumps is relatively mature, however, few results are obtained on the necessary and sufficient conditions for establishing the stability of the true solution and the numerical solution. This project firstly discusses the numerical simulation of the G-Brownian motion, then presents the numerical simulation of stochastic differential equations driven by G-Brownian motion, and analyzes its convergence and stability. In addition, the necessary and sufficient conditions between the stability of the true solutions and the stability of the numerical solutions to stochastic differential equations driven by G-Brownian motion and stochastic differential equations with Poisson jumps are established. Therefore, we can study the stability of stochastic differential equations by some numerical methods instead of constructing Lyapunov functions. And we expect that this project can make a new breakthrough.