| In this thesis, we investigate the properties of solution to the stochastic differential equation driven by fraction Brownian motion with Hurst parameter H > 1/4. In particular, we study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Holder path with exponent beta > 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 >1/2. 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 < H < 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 Hormander's type condition. |
