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System Control and Rough Paths$
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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

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PATH INTEGRATION ALONG ROUGH PATHS

PATH INTEGRATION ALONG ROUGH PATHS

Chapter:
(p.110) 5 PATH INTEGRATION ALONG ROUGH PATHS
Source:
System Control and Rough Paths
Author(s):

Terry Lyons

Zhongmin Qian

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506485.003.0005

This chapter establishes an integration theory for rough paths. The key result is the existence of the integral of a Lipschitz one-form against a rough path. In particular, this gives a change of variable formula, and allows one to see that the notion of a rough path is a geometric concept, stable under smooth change of variable. A key technical point is the use of the notion of Lipschitz, to be found in Stein's book ‘Singular integrals’, which allows for functions to be Lipschitz of any degree. The basic integration result requires the one-form to be Lipschitz of degree > p-1, in order to integrate to against all p-rough paths.

Keywords:   Lipschitz one-form, rough path, stochastic differential equation, Chen iterated integral, Itô functional

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