Abstract
Rough Path theory is currently formulated in p-variation 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 fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (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 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 Holder topology as immediate corollary.
The author acknowledges financial support by the Austrian Academy of Science.
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References
G. BEN AROUS, M. GRÂDINARU, AND M. LEDOUX, Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincare Probab. Statist. 30(3): 415–436 (1994).
N. IKEDA AND S. WATANABE, SDEs and diffusion processes. North Holland (1981, 89).
A. LEJAY, ntroduction to Rough Paths, www.iecn.u-nancy.fr/∼lejay/ rough.html, to appear in Sém. de Proba., LNM, Springer.
T. LYONS, Differenti al Equations Driven by Rough Signals, Math. Res. Letters. 1: 451–464 (1994).
T. LYONS, Int erp retat ion and Solution of ODEs Driven by Rough Signals, Proc. Sypmosia Pure Math. 57: 115–128 (1995).
T. LYONS, Different ial Equations driven by rough signals, Revist a Matemati ca Iberoamericana. 14(2): 215–310 (1998).
T. LYONS AND Z. QIAN, Calculus of Variation for Multiplicative Functionals, New Trends in Sto chastic Anal ysis, 348–374 Charingworth 1994. World Scientific, River Edge, NJ (1997).
T. LYONS AND Z. QIAN, System Control and Rough Paths, Oxford University Press (2002).
M. LEDOUX, Z. QIAN, AND T. ZHANG, Large deviations and support theorem for diffusion processes via Rough Paths, Stoch. Proc. Appl. 102(2): 265–283 (2002).
P. MALLIAVIN, Stochastic Analys is, Springer (1997).
A. MILLET AND M. SANZ-SOLE, A simple proof of the support th eorem for diffusion pro cesses, LNM 1583, pp. 36–48.
D. REVUZ AND M. YOR, Continuous Martingales and Brownian Moti on, Springer (1999).
E. SIPPILAINEN, A pathwise view of solutions to SDEs, unpublished PhD-thesis Univ. of Edinburgh (1993).
D.W. STROOCK AND S. TANIGUCHI, Diffusions as int egral curv es, or Stratonovich without !to integration, Birkhäuser, Prog. Probab. 34: 333–369 (1994).
D.W. STROOCK AND S.R.S. VARADHAN, On the support of diffusion processes with applicat ion to the strong maximum principle. Proc. 6th Berkeley Symp. Math. Statist. Prob. III. Univ. California Press, 333–359 (1972).
L.C. YOUNG, An inequality of Holder type, connected with Stieltjies integration. Acta Math. 67: 251–82.
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Friz, P.K. (2005). Continuity of the Itô-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm. In: Waymire, E.C., Duan, J. (eds) Probability and Partial Differential Equations in Modern Applied Mathematics. The IMA Volumes in Mathematics and its Applications, vol 140. Springer, New York, NY. https://doi.org/10.1007/978-0-387-29371-4_8
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DOI: https://doi.org/10.1007/978-0-387-29371-4_8
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