Some aspects of stochastic differential equation driven by fractional Brownian motion

In this thesis, we investigate the properties of solution to the stochastic differential equation driven by fraction Brownian motion with Hurst parameter H \u3e 1/4. In particular, we study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent β \u3e 1/2. We derive a convergence criterion that enable us to write the solution as an infinite sum of iterated integrals on a nonempty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter H \u3e1/2. We also study the convergence in L 2 of the stochastic Taylor expansion by using L 2 estimates of iterated integrals and Borel-Cantelli type arguments. With the rough path analysis tool, we extend our results to include the case 1/4 \u3c H \u3c 1/4. The regularization estimates we obtain generalize to the fractional Brownian previous results by Kusuoka and Strook and can be seen as a quantitative version of the existence of smooth densities under Hörmander\u27s type condition