满足交换性条件的随机微分方程高阶分裂方法
High order splitting methods for SDEs satisfying a commutativity condition
AI总结：
本研究提出用分段线性路径替换SDE驱动信号的新视角，构建满足交换性条件的SDE高阶分裂方法，证明其强收敛性，实验显示收敛速率达$O(h^{3/2})$，性能优于现有方法。
摘要：
本文提出了一种全新的简易思路，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，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（即布朗运动与时间），由此得到一系列常微分方程（ODE），对这些ODE进行离散化便可生成数值格式。这种理解分裂方法的新视角受粗糙路径理论启发，但并未用到该理论。我们证明，当驱动的分段线性路径匹配布朗运动的某些迭代随机积分时，即可得到高阶分裂方法。我们提出了一套证明这类近似强收敛性的通用方法体系，与Milstein和Tretyakov的通用框架类似：一旦得到分裂方法的局部误差估计，即可推导得到全局收敛速率。该方法可直接应用于未来SDE分裂方法的研究。通过将最新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，我们针对满足某一交换性条件的SDE提出了多种高阶分裂方法。我们的实验涵盖Cox-Ingersoll-Ross模型以及加性噪声SDE（含噪声非简谐振子、随机FitzHugh-Nagumo模型、欠阻尼朗之万动力学），结果表明新分裂方法的收敛速率为$O(h^{3/2})$，性能优于现有文献中已提出的格式。
英文摘要：
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.
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