In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

在这项工作中，开发了一种用于随机微分方程（SDE）的连续阶段随机Runge–Kutta（CSSRK）方法。首先，通过随机B系列和多色根树理论建立了这些方法的一般顺序理论。然后将提出的CSSRK方法应用于三种特殊的SDE，并推导了相应的订购条件。特别是，对于单个被积分SDE和具有加性噪声的SDE，我们构造了一些特定的高阶CSSRK方法。此外，事实证明，借助不同的数值正交公式，CSSRK方法可以生成具有相同顺序的相应随机Runge-Kutta（SRK）方法。因此，产生了一些有效的SRK方法。最后，通过一些数值实验证明了这些理论结果。