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

In this note we construct solutions to rough differential equations $\mathrm{d}Y=f(Y)\mathrm{d}X$ with a driver $X\in C^{\alpha }([0,T];{\mathbb{R}}^d)$, $\frac{1}{3}<\alpha \le \frac{1}{2}$, 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.