Jump to ContentJump to Main Navigation
System Control and Rough Paths$
Users without a subscription are not able to see the full content.

Terry Lyons and Zhongmin Qian

Print publication date: 2002

Print ISBN-13: 9780198506485

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506485.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 16 January 2022

BROWNIAN ROUGH PATHS

BROWNIAN ROUGH PATHS

Chapter:
(p.61) 4 BROWNIAN ROUGH PATHS
Source:
System Control and Rough Paths
Author(s):

Terry Lyons

Zhongmin Qian

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506485.003.0004

This chapter proves that the dyadic polygonal approximations to a Brownian paths, with probability 1, a Cauchy sequence of rough paths, and in this way, the chapter proves a result first established in its simplest form by E. Sipilainen, that Brownian motion, with Lévy area, can be regarded as a rough path. This basic result can be extended to Brownian motion in Banach spaces and to fractional Brownian motions with Hurst index < 4. Since the notion of rough paths depends on computing iterated integrals, and these live in tensor product spaces, any discussion of rough paths in infinite dimensional contexts requires knowledge of different tensor norms. Brownian motion in a Banach space is always a rough path in the injective tensor norm, and normally a rough path in the projective tensor norm. Indeed, it is an open problem to construct an example where the Lévy area did not exist in the projective norm.

Keywords:   Brownian paths, Cauchy sequence, Banach space, Lévy area, Chen iterated integral

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .