Euler Estimates of Rough Differential Equations

We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. As application, we can control moments of solutions to rough path differential equations (RDEs) driven by random rough paths with sufficient integrability and have a criteria for L^q - convergence in the Universal Limit Theorem. We also obtain Azencott type estimates and asymptotic expansions for random RDE solution. When specialized to RDEs driven by Enhanced Brownian motion, we (mildly) improve classic estimates for diffusions in the small time limit.