The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Itô-Taylor schemes for RODEs with Itô noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.

中文翻译：

---

随机常微分方程数值方法的均方收敛

这项工作的目的是分析随机常微分方程（RODE）的数值方案的均方收敛速度。首先，建立单步显式逼近的全局均方收敛与局部均方收敛阶之间的关系。然后，分别研究了一般RODE的RODE-Taylor方案，仿射噪声的RODE的Affine-RODE-Taylor方案和Itô噪声的RODE的Itô-Taylor方案的全局均方收敛速度。通过数值实验证明了理论收敛结果。