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

TL;DR

This paper derives a precise Taylor remainder estimate for differential equations driven by weakly geometric $ ext{Pi}$-rough paths, improving understanding of solution approximations in rough path theory.

Contribution

It provides a novel remainder estimate for Taylor expansions in rough differential equations driven by $ ext{Pi}$-rough paths, including a refined factorial decay component under certain conditions.

Findings

Remainder estimates are of the correct order, comparable to the next Taylor term.

A refined estimate with factorial decay is obtained when specific conditions on $ ext{Pi}$ are met.

The results enhance the accuracy of Taylor approximations in rough path driven differential equations.

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) $, $p_{i}\geq 1$. 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.