  In this paper we establish a Taylor-like expansion in the context of the
rough path theory for a family of It ^{o} maps indexed by a small parameter. We
treat not only the case that the roughness $p$ satisfies $[p]=2$, but also the
case that $[p] ge 3$. As an application, we discuss the Laplace asymptotics for
It^{o} functionals of Brownian rough paths.
