Continuity of the Itô-map for Hölder Rough Paths with Applications to the Support Theorem in Hölder Norm
نویسنده
چکیده
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 Hölder metric ρ can be used. Based on fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Itô-map (between Rough Path spaces equipped with ρ). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t. ρ. In combination with our Hölder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called 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 Stroock-Varadhan Support Theorem in Hölder topology as immediate corollary.
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