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

Abstract

We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $\Pi$-rough paths for $\Pi =\left(p_1,\cdots ,p_k\right)$, $p_i\ge 1$. When there exists $p\ge 1$ such that $p_i=p{k}_i^{-1}$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.