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Stochastic analysis, rough path analysis and fractional Brownian motions. (English) Zbl 1047.60029
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. L. Decreusefond and 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.

##### MSC:
 60G15 Gaussian processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 60G22 Fractional processes, including fractional Brownian motion 60L20 Rough paths
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