系数不连续的随机微分方程及其数值分析

对漂移项系数不连续、扩散项系数退化的随机微分方程，本项目将开展以下方面研究：分析数值方法的强收敛性和弱收敛性，建立单步数值方法的基本收敛性定理，研究高阶数值方法的构造及实现，以期获得高效数值方法；研究方程本身的稳定性和数值方法的稳定性，稳定性包括关于初始值的稳定性、长时间稳定性（如均方稳定性等）；在降低对间断面的光滑性要求下，研究方程解的存在唯一性及数值解对系数正则性的依赖关系；研究方程本身的不变测度和遍历性，构造保持遍历性的数值方法，并分析数值方法不变测度的收敛性及收敛阶，进一步将上述遍历性研究推广到随机偏微分方程及其数值格式。本项目所获结果将丰富和发展随机微分方程的数值算法理论，并且在自动控制、金融衍生品等领域具有广泛的应用前景。

For stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion, this project will carry out the following researches: We will study strong convergence and weak convergence of numerical methods, establish a fundamental theorem of convergence of one step methods, design higher order numerical methods and investigate how to implement them effectively so as to obtain efficient numerical schemes. We will investigate stability of the SDEs and numerical methods (stabilities include the stability with respect to initial values and long-time stability, such as mean square stability and so on). In the case of lower regularity of the discontinuous hypersurface, we will study existence and uniqueness of the solutions of the SDEs and find out how the numerical solutions depends on the regularity on the coefficients. We will investigate invariant measure and ergodicity of the SDEs and numerical methods, study convergence and convergence rate of invariant measure of the numerical methods. Further, we extend the study on ergodicity to stochastic partial differential equations and numerical methods. The results of this project will enrich and develop numerical analysis theory of stochastic differential equations, and have broad application prospects in the fields of automatic control and financial derivatives and so on.