Convergence rate in the splitting-up method for rough differential equations

TL;DR

This paper develops a splitting-up numerical scheme for rough differential equations driven by Hölder continuous signals with exponent between 1/3 and 1/2, proving convergence and providing a rate of convergence.

Contribution

It introduces a novel splitting-up method for rough differential equations and establishes convergence with a new proof technique, along with a convergence rate.

Findings

The scheme converges to the solution in the sense of Davie.

A specific rate of convergence is established.

The method applies to rough signals with Hölder exponent between 1/3 and 1/2.

Abstract

In this note we construct solutions to rough differential equations ${\rm d} Y = f(Y) \,{\rm d} X$ with a driver $X \in C^\alpha([0,T];\mathbb{R}^d)$, $\frac13 < \alpha \le \frac12$, using a splitting-up scheme. We show convergence of our scheme to solutions in the sense of Davie by a new argument and give a rate of convergence.