Taylor estimate for differential equations driven by $\mathrm{\Pi }$-rough paths

We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $\mathrm{\Pi }$-rough paths for $\mathrm{\Pi }=(p_1,\cdots ,p_k)$, $p_i\ge 1$. When there exists $p\ge 1$ such that p_{i}=pk_{i}^{-1}\for some $k_i\in \{1,\dots ,\left[p\right]\}$, we obtain a refined Taylor remainder estimate that contains a factorial decay component. The remainder estimates are in the right order as they are comparable to the next term in the Taylor expansion.