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：本文提出一种简单的新方法来构建并证明一类广泛随机微分方程（SDEs）分裂方法的收敛性，包括加性噪声、对角噪声和标量噪声类型。核心思想是将分裂方法视为对SDE驱动信号（即布朗运动和时间）的替换：通过分段线性路径生成一系列常微分方程（ODEs）——可经离散化得到数值格式。这种理解分裂方法的新思路受粗糙路径理论启发，但未直接使用该理论。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，可获得高阶分裂方法。我们提出一种通用的证明方法论——类似于Milstein和Tretyakov的框架——来建立这些逼近的强收敛性：一旦获得分裂方法的局部误差估计，便能推导出全局收敛速率。该方法可便捷地应用于未来SDE分裂方法的研究。通过将近期发展的布朗运动迭代积分逼近技术融入分段线性路径，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模型和加性噪声SDEs（有噪声非谐振荡器、随机FitzHugh-Nagumo模型、欠阻尼Langevin动力学）的实验中，新分裂方法展现出$O(h^{3/2})$的收敛速率，并优于文献中先前提出的方案。