A Note on the Notion of Geometric Rough Paths
Functional Analysis
2007-05-23 v1 Probability
Abstract
We use simple sub-Riemannian techniques to prove that an arbitrary geometric p-rough path in the sense of Lyons (98) is the limit in sup-norm of a sequence of canonically lifted smooth paths, which are uniformly bounded in p-variation, clarifying the two different definitions of a geometric p-rough path, Lyons (98), Lyons/Qian (02). Our proofs are based on fine estimates in terms of control functions and are sufficiently general to include the case of Hoelder- and modulus-type regularity, Friz/Victoir (03). This allows us to extend a few classical results on Hoelder-spaces (Ciesielski, Musielak/Semadeni) and p-variation spaces (Wiener,Dudley) to the non-commutative setting necessary for the theory of rough paths.
Keywords
Cite
@article{arxiv.math/0403115,
title = {A Note on the Notion of Geometric Rough Paths},
author = {Peter Friz and Nicolas Victoir},
journal= {arXiv preprint arXiv:math/0403115},
year = {2007}
}