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:在本文中,我们引入了一种新的简单方法,为大型类随机差异方程式(SDEs)开发和建立分解方法的趋同,包括添加式、对角和卡路里噪声类型。核心理念是将分解方法视为SDE驱动信号的替代,即布朗尼运动和时间,使用一条小片线性路径,产生一系列可分解的数值-美元,以产生数字方案。这种新的理解分解方法受粗路径理论的启发,但没有使用。我们展示,当驱动片断线性路径与布朗运动的某些迭接轨、对面和卡路里混杂音组合相匹配时,就可以找到一种高顺序分解方法。我们提出了一个一般的证明方法,以建立这些近似Milstein和Tretyakov总框架的近似一致。这就是,一旦获得对分解方法的局部误差估计,然后是全球趋同率。然后可以在SDE分解方法的未来研究中应用这一方法。我们通过最近开发的纸质平流- 平流C 将某种直径直径路方法纳入我们不断推进的Sral-rodeal-rodeal rodeal rodeal 。