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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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System Control and Rough Paths

Terry Lyons

Zhongmin Qian

Oxford University Press

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

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