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A Milstein-type scheme was proposed to improve the rate of convergence of itsapproximation of the solution to a stochastic differential equation driven by avector of continuous semimartingales. A necessary and sufficient condition wasprovided for this rate to be $1/n$ when the SDE is driven by a vector ofcontinuous local martingales, or continuous semimartingales under an additionalassumption on their finite variation part. The asymptotic behavior (weakconvergence) of the normalized error processes was also studied.

Asymptotic error for the Milstein scheme for SDEs driven by continuous semimartingales