Probability and Partial Differential Equations in Modern Applied Mathematics
Volume 140 of the series The IMA Volumes in Mathematics and its Applications pp 117135
Continuity of the ItôMap for Holder Rough Paths with Applications to the Support Theorem in Holder Norm
 Peter K. FrizAffiliated withCourant Institute, NYU
Abstract
Rough Path theory is currently formulated in pvariation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric π can be used. Based on fineestimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitzcontinuity of the Itornap (between Rough Path spaces equipped with π). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t. π. In combination with our Holder ULT this allows sharper results than the pvariation theory. Also, our formulation avoids the socalled control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the StroockVaradhan Support Theorem in Holder topology as immediate corollary.
Key words
Rough Path theory Itômap Universal Limit Theorem pvariation vs. Hölder regularity Support TheoremAMS(MOS) subject classifications
60Gxx Title
 Continuity of the ItôMap for Holder Rough Paths with Applications to the Support Theorem in Holder Norm
 Book Title
 Probability and Partial Differential Equations in Modern Applied Mathematics
 Pages
 pp 117135
 Copyright
 2005
 DOI
 10.1007/9780387293714_8
 Print ISBN
 9780387258799
 Online ISBN
 9780387293714
 Series Title
 The IMA Volumes in Mathematics and its Applications
 Series Volume
 140
 Series ISSN
 09406573
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media, Inc.
 Additional Links
 Topics
 Keywords

 Rough Path theory
 Itômap
 Universal Limit Theorem
 pvariation vs. Hölder regularity
 Support Theorem
 60Gxx
 Industry Sectors
 eBook Packages
 Editors

 Edward C. Waymire ^{(2)}
 Jinqiao Duan ^{(3)}
 Editor Affiliations

 2. Department of Mathematics, Oregon State University
 3. Department of Applied Mathematics, Illionis Institute of Technology
 Authors

 Peter K. Friz ^{(4)}
 Author Affiliations

 4. Courant Institute, NYU, New York, NY, 10012
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