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来建立和证明满足大范围随机微分方程（包括加性噪声、对角噪声和标量噪声类型）的分裂方法的收敛性。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，通过分段线性轨迹得到一系列常微分方程，可离散化以生成数值方案。这种理解分裂方法的新方式受粗糙路径理论的启发，但并未使用该理论。我们证明，当分段线性驱动轨迹匹配布朗运动的特定迭代随机积分时，可获得高阶分裂方法。我们提出了一种类似于米尔斯坦和特列季亚科夫通用框架的通用证明方法，用于建立这些近似值的强收敛性。即一旦获得分裂方法的局部误差估计，则全局收敛性随之而来。该方法可快速应用于未来随机微分方程分裂方法的研究。通过将近期开发的布朗运动迭代积分近似值整合到这些分段线性路径中，我们提出了几种满足特定交换条件的随机微分方程的高阶分裂方法。在我们的实验（包括考克斯-英格索尔-罗斯模型、加性噪声随机微分方程（含噪声非调和振子、随机菲茨休-南云模型、欠阻尼朗之万动力学）中，新分裂方法展现出$O(h^{3/2})$的收敛速率，且优于文献中先前提出的方案。