A uniform estimate for rough paths

TL;DR

This paper extends the classical rough paths continuity theorem, showing that if the initial levels of two rough paths are close uniformly, then all their iterated integrals are close, with applications to signature estimation and Gaussian rough paths convergence.

Contribution

It provides a new uniform estimate linking initial path closeness to all levels of rough path iterated integrals, enhancing understanding of rough path stability.

Findings

Two $p$-rough paths are close in all levels if initial levels are uniformly close.

Applications include estimating signature differences of close paths.

Results provide convergence rates for Gaussian rough paths.

Abstract

We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications include estimation of the difference of the signatures of two uniformly close paths and convergence rates of Gaussian rough paths.