Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations
Abstract
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∈(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 r0>0, the simplified method has global error O(hmin{r0,(p+1)α−1}). Numerical experiments with fractional Brownian controlled drivers illustrate the resulting convergence behaviour.
Related
- Unconditionally L2-stable explicit--implicit--null filtered Runge--Kutta methods for general nonlinear degenerate parabolic equations
- Numerical methods for nonlinear stochastic differential equations driven by subdiffusion
- Random dynamical systems generated by rough differential equations driven by controlled rough paths
- Toward Spectral Deferred Correction Methods for Stochastic Differential Equations
- Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
BibTeX
@misc{an2026controlled,
title = {{Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations}},
author = {Xinyuan An and Xingya Fan and Xing Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14196},
eprint = {2609.14196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14196},
}
RIS
TY - UNPB
TI - Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations
AU - Xinyuan An
AU - Xingya Fan
AU - Xing Gao
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14196
UR - https://arxiv.org/abs/2609.14196
AB - 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∈(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.
ER -