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

A stochastic Taylor-like expansion in the rough path theory