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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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UNIVERSAL LIMIT THEOREM

UNIVERSAL LIMIT THEOREM

Chapter:
(p.148) 6 UNIVERSAL LIMIT THEOREM
Source:
System Control and Rough Paths
Author(s):

Terry Lyons

Zhongmin Qian

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506485.003.0006

The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential equations controlled by a p-rough path has a unique meaning for p-rough paths, providing the system of equations has a Lipschitz smoothness > p. In fact, the proof of this result also shows that the response of the Itô functional to the driving signal is continuous in the p-rough path metric. The proof is a difficult iterative process, but because the bounds are uniform, one sees that the iterations converge uniformly, and so the limit is continuous.

Keywords:   Chen iterated integral, universal limit theorem, stochastic differential equation, p-rough path, Itô functional

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