### Fractional Brownian motion and the Markov property.

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We present the rough path theory introduced by Lyons, using the swewing lemma of Feyel and de Lapradelle.

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus...

A stochastic “Fubini” lemma and an approximation theorem for integrals on the plane are used to produce a simulation algorithm for an anisotropic fractional Brownian sheet. The convergence rate is given. These results are valuable for any value of the Hurst parameters $({\alpha}_{1},{\alpha}_{2})\in {]0,1[}^{2},{\alpha}_{i}\ne \frac{1}{2}.$ Finally, the approximation process is iterative on the quarter plane ${\mathbb{R}}_{+}^{2}.$ A sample of such simulations can be used to test estimators of the parameters = 1,2.

We propose some construction of enhanced Gaussian processes using Karhunen-Loeve expansion. We obtain a characterization and some criterion of existence and uniqueness. Using rough-path theory, we derive some Wong-Zakai Theorem.

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