In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretised to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.

翻译：本文提出了一种新的简单方法，用于构造并证明一类广泛随机微分方程（SDEs）分裂方法的收敛性，涵盖加性噪声、对角噪声和标量噪声类型。核心思想是将分裂方法视为用分段线性路径替换SDE的驱动信号（即布朗运动和时间的函数），从而生成一系列常微分方程（ODEs），这些方程可经离散化以得到数值格式。这种理解分裂方法的新思路受粗糙路径理论启发，但并未实际应用该理论。我们证明，当驱动分段线性路径匹配布朗运动的某些迭代随机积分时，便可获得高阶分裂方法。本文提出了一种通用的证明方法学，用于建立这些近似方法的强收敛性，其框架类似Milstein和Tretyakov的一般框架：一旦获得分裂方法的局部误差估计，即可推导出全局收敛阶次。该方法可便捷地应用于未来SDE分裂方法的研究中。通过将近期发展的布朗运动迭代积分近似方法融入这些分段线性路径，我们针对满足特定交换性条件的SDEs提出了若干高阶分裂方法。在包含Cox-Ingersoll-Ross模型及加性噪声SDEs（含噪声非谐振子、随机FitzHugh-Nagumo模型、欠阻尼Langevin动力学）的实验中，新型分裂方法展现出$O(h^{3/2})$的收敛阶，并优于文献中先前提出的方案。