随机微分方程广泛应用于模拟物理学、经济学、生物学等诸多领域中的问题。因为大部分随机微分方程无法求出其精确解，对于随机微分方程数值方法的研究是非常必要的。能够保持原系统某些特定结构的数值方法被称为保结构算法，保结构算法具有良好的性质，尤其在长时间数值模拟中具有不可比拟的优势。本项目致力于研究随机微分方程的连续级Runge-Kutta型保结构算法。主要考虑以下两个方面：1）将求解确定性微分方程的连续级Runge-Kutta方法推广到随机情形，研究保持辛结构或守恒量的连续级随机Runge-Kutta方法；2）将连续级随机Runge-Kutta方法进一步推广到分块情形，研究保持辛结构或守恒量的连续级随机分块Runge-Kutta方法。本项目不仅可以丰富随机微分方程的保结构算法理论，还可以为一些实际问题提供算法支持。

Stochastic differential equations are widely used to model problems in many fields such as physics, economics, biology, and so on. Since most stochastic differential equations cannot be solved exactly, the study of numerical methods for stochastic differential equations is very necessary. Numerical methods that can preserve certain structure of the original systems are called structure-preserving algorithms, which have good properties and overwhelming advantages especially in a long-term numerical simulation. This project devotes to studying structure-preserving algorithms of continuous-stage Runge-Kutta type for stochastic differential equations. The following two aspects are mainly considered: 1) To extend the continuous-stage Runge-Kutta methods for ordinary differential equations to the stochastic case and study the symplectic or invariant-preserving continuous-stage stochastic Runge-Kutta methods. 2) To further extend the continuous-stage stochastic Runge-Kutta methods to the partitioned case and study the symplectic or invariant-preserving continuous-stage stochastic partitioned Runge-Kutta methods. This project can not only enrich the theory of structure-preserving algorithms for stochastic differential equations, but also provide algorithm support for some practical problems.