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We derive an asymptotic expansion for the quadratic variation of a stochasticprocess satisfying a stochastic differential equation driven by a fractionalBrownian motion, based on the theory of asymptotic expansion of Skorohodintegrals converging to a mixed normal limit. In order to apply the generaltheory, it is necessary to estimate functionals that are a randomly weightedsum of products of multiple integrals of the fractional Brownian motion, inexpanding the quadratic variation and identifying the limit random symbols. Toovercome the difficulty, we introduce two types of exponents by means of the"weighted graphs" capturing the structure of the sum in the functional, andinvestigate how the exponents change by the action of the Malliavin derivativeand its projection.

Order estimate of functionals related to fractional Brownian motion and  asymptotic expansion of the quadratic variation of fractional stochastic  differential equation