Continuity of the Itô-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm

  • Peter K. Friz

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.

Key words

Rough Path theory Itô-map Universal Limit Theorem p-variation vs. Hölder regularity Support Theorem 

AMS(MOS) subject classifications

60Gxx 

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References

  1. [BGL]
    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).Google Scholar
  2. [IW]
    N. IKEDA AND S. WATANABE, SDEs and diffusion processes. North Holland (1981, 89).Google Scholar
  3. [Le]
    A. LEJAY, ntroduction to Rough Paths, www.iecn.u-nancy.fr/∼lejay/ rough.html, to appear in Sém. de Proba., LNM, Springer.Google Scholar
  4. [L94]
    T. LYONS, Differenti al Equations Driven by Rough Signals, Math. Res. Letters. 1: 451–464 (1994).Google Scholar
  5. [L95]
    T. LYONS, Int erp retat ion and Solution of ODEs Driven by Rough Signals, Proc. Sypmosia Pure Math. 57: 115–128 (1995).Google Scholar
  6. [L98]
    T. LYONS, Different ial Equations driven by rough signals, Revist a Matemati ca Iberoamericana. 14(2): 215–310 (1998).Google Scholar
  7. [LQ97]
    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).Google Scholar
  8. [LQ]
    T. LYONS AND Z. QIAN, System Control and Rough Paths, Oxford University Press (2002).Google Scholar
  9. [LQZ]
    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).Google Scholar
  10. [M]
    P. MALLIAVIN, Stochastic Analys is, Springer (1997).Google Scholar
  11. [MS]
    A. MILLET AND M. SANZ-SOLE, A simple proof of the support th eorem for diffusion pro cesses, LNM 1583, pp. 36–48.Google Scholar
  12. [RY]
    D. REVUZ AND M. YOR, Continuous Martingales and Brownian Moti on, Springer (1999).Google Scholar
  13. [SI]
    E. SIPPILAINEN, A pathwise view of solutions to SDEs, unpublished PhD-thesis Univ. of Edinburgh (1993).Google Scholar
  14. [ST]
    D.W. STROOCK AND S. TANIGUCHI, Diffusions as int egral curv es, or Stratonovich without !to integration, Birkhäuser, Prog. Probab. 34: 333–369 (1994).Google Scholar
  15. [SV]
    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).Google Scholar
  16. [Y]
    L.C. YOUNG, An inequality of Holder type, connected with Stieltjies integration. Acta Math. 67: 251–82.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Peter K. Friz
    • 1
  1. 1.Courant InstituteNYUNew York

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