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: 21 January 2022



System Control and Rough Paths

Terry Lyons

Zhongmin Qian

Oxford University Press

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is developed and an explicit answer is given to the question. Exactly what information should I extract from the driving stimulus or noise in order to accurately predict the response? The notions of a controlled system of Chen's iterated integral are introduced. The main notion in this book is the concept of a rough path. Almost all paths that one encounters in everyday life are only described approximately. Newton observed that a smooth path is actually quite well approximated by its chords. If one wants to describe a path γ over a short time interval from s to t, then it is enough to evaluate γ at these two times and consider the approximation that comes from replacing γ by the straight line with the same increment. This approach is not adequate if the control or path γ is oscillatory on the scale witnessed by the times s and t. If the path γ represented a text, then the chord is simply a word count. It turns out that a better description, one which takes into account the order of the events represented by γ, can be achieved by a description of γ that involves its first few Chen iterated integrals.

Keywords:   Chen iterated integral, rough path, smooth path, Newton, two-dimensional noise

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 .