  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.
