Sensitivity of rough differential equations∗

In the context of rough paths theory, we study the regularity of the Itô map with respect to the starting point, the coefficients and perturbation of the driving rough paths. In particular, we show that the Itô map is differentiable with a Hölder or Lipschitz continuous derivatives. With respect to the current literature on the subject, our proof rely on perturbation of linear Rough Differential Equations and our approach under the most general hypotheses on the driving path and on the vector field.