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

We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $\Pi$-rough paths for $\Pi =\left( p_{1},\cdots ,p_{k}\right)$, $p_{i}\geq 1$. When there exists $p\geq 1$ such that $p_{i}=pk_{i}^{-1}\$for some $k_{i}\in \left\{ 1,\dots , \left[ p\right] \right\}$, 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.