Probability Theory and Related Fields

, Volume 122, Issue 1, pp 108-140

First online:

Stochastic analysis, rough path analysis and fractional Brownian motions

  • Laure CoutinAffiliated withCNRS and Université Paul-Sabatier, Laboratoire de Statistique et Probabilités, 118 Route de Narbonne, 31062 Toulouse, France. e-mail: coutin@cict.fr; qian@cict.fr
  • , Zhongmin QianAffiliated withCNRS and Université Paul-Sabatier, Laboratoire de Statistique et Probabilités, 118 Route de Narbonne, 31062 Toulouse, France. e-mail: coutin@cict.fr; qian@cict.fr

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Abstract.

 In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition.