Stochastic analysis, rough path analysis and fractional Brownian motions. (Q1862500)

This paper contains the proofs of the results announced in [C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 1, 75--80 (2000; Zbl 0981.60040)] and deals with rough path theory at the third level for fractional Brownian motion with Hurst parameter strictly greater than 1/4. A dyadic approximation theorem for the enhanced fractional Brownian path is shown, with respect to the \(p\)-variation distance. Moreover, an explicit kernel representation of the two last components of the enhanced path is given [the first one being classical -- see e.g. \textit{L. Decreusefond} and \textit{A. S. Üstünel}, Potential Anal. 10, No. 2, 177--214 (1999; Zbl 0924.60034)]. Last, the authors prove a Wong-Zakai approximation theorem in \(p\)-variation for enhanced SDEs driven by fBm with Hurst parameter greater than 1/4, and prove that the latter define a smooth flow of diffeomorphisms. Notice that this last property is not a straightforward application of Lyons' universality theorem.