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 discretized 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，用于构建并证明一大类随机微分方程（SDE）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。其核心思想是将分裂方法视为对随机微分方程驱动信号（即布朗运动和时间）的替代，通过分段线性路径生成一系列常微分方程，这些方程可离散化以产生数值方案。这种理解分裂方法的新途径受粗路径理论启发，但并未直接使用该理论。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，可以得到高阶分裂方法。我们提出了一种通用的证明方法，用于建立这些近似方法的强收敛性，该方法类似于Milstein和Tretyakov的通用框架。即一旦获得分裂方法的局部误差估计，即可推导出全局收敛率。这种方法可方便地应用于未来SDE分裂方法的研究。通过将最近开发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律条件的SDE提出若干高阶分裂方法。在我们的实验中（包括Cox-Ingersoll-Ross模型和加性噪声SDE：含噪声非简谐振子、随机FitzHugh-Nagumo模型、欠阻尼朗之万动力学），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。