A uniform estimate for rough paths

It is well known that for two p-rough paths, if their first ⌊p⌋ levels of iterated integrals are close in p-variation sense, then all levels of their iterated integrals are close. In this paper, we prove that a similar result holds for the paths provided the first ⌊p⌋ terms are close in a ‘uniform’ sense. The estimate is explicit, dimension free, and only involves the p-variation of two paths and the ‘uniform’ distance between the first ⌊p⌋ terms. Applications include estimation of the difference of the signatures of two uniformly close paths (Lyons and Xu, 2011 [6], ), and convergence rates for Gaussian rough paths (Riedel and Xu, 2012 [7]).