受控B-级数、受控Runge-Kutta方法及受控粗糙微分方程
Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations
浏览论文内容
中文总结 AI 辅助
本文提出受控B-级数和受控Runge-Kutta方法，用于求解受控驱动的粗糙微分方程，推导了树阶条件并给出误差估计，简化方法仅需驱动增量，数值实验验证了收敛性。
中文摘要 AI 辅助
我们将受控B-级数和受控Runge-Kutta方法发展为经典B-级数和Runge-Kutta理论对受控驱动的粗糙微分方程的扩展。驱动信号是由底层α-Hölder阶2粗糙路径控制的路径，其中α∈(1/3,1/2]，这出现在一个粗糙系统的输出驱动另一个系统时。受控B-级数保留了经典的根树和初等微分结构，同时通过受控粗糙积分递归定义依赖于驱动的系数。我们建立了精确解及其受控Runge-Kutta逼近的有限阶展开，并推导了相应的基于树的阶条件。在适当的光滑性、可解性、稳定性和有界性假设下，通过树阶p的匹配产生局部误差O(|t-s|^{(p+1)α})和全局误差O(h^{(p+1)α-1})，前提是(p+1)α>1。当受控驱动是参考路径本身时，恢复标准粗糙微分方程公式，而经典时间驱动情形恢复通常的B-级数和Runge-Kutta方法。我们进一步构造了一种仅使用受控驱动增量的简化方法。其分段线性插值的分段规范提升给出显式树系数，并将三阶条件简化为经典条件。假设Wong-Zakai解逼近速率r0>0，简化方法的全局误差为O(h^{min{r0,(p+1)α-1}})。使用分数布朗运动受控驱动的数值实验说明了由此产生的收敛行为。
英文摘要
We develop controlled B-series and controlled Runge-Kutta methods as extensions of classical B-series and Runge-Kutta theory to controlled-driven rough differential equations. The driving signal is a path controlled by an underlying $α$-Hölder step-$2$ rough path, with $α\in(1/3,1/2]$, as arises when the output of one rough system drives another. Controlled B-series retain the classical rooted-tree and elementary-differential structure, while incorporating driver-dependent coefficients defined recursively through controlled rough integration. We establish finite-order expansions of the exact solution and its controlled Runge-Kutta approximation, and derive corresponding tree-based order conditions. Under suitable smoothness, solvability, stability, and boundedness assumptions, matching through tree order $p$ yields local error $O(|t-s|^{(p+1)α})$ and global error $O(h^{(p+1)α-1})$, provided $(p+1)α>1$. The standard rough differential equation formulation is recovered when the controlled driver is the reference path itself, and the classical time-driven case recovers the usual B-series and Runge-Kutta methods. We further construct a simplified method using only increments of the controlled driver. Segmentwise canonical lifting of its piecewise linear interpolation gives explicit tree coefficients and reduces the third-order conditions to the classical ones. Assuming a Wong-Zakai solution-approximation rate $r_0>0$, the simplified method has global error $O(h^{\min\{r_0,(p+1)α-1\}})$. Numerical experiments with fractional Brownian controlled drivers illustrate the resulting convergence behaviour.
发表机构
- Lanzhou University（兰州大学）
- Xinjiang University（新疆大学）
- Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics（甘肃省数学与统计基础学科研究中心）
机构由 AI 辅助整理，请以论文原文为准。
