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We study the $L^p$ rate of convergence of the Milstein scheme for SDEs whenthe drift coefficients possess only H\"older regularity. If the diffusion iselliptic and sufficiently regular, we obtain rates consistent with the additivecase. The proof relies on regularisation by noise techniques, particularlystochastic sewing, which in turn requires (at least asymptotically) sharpestimates on the law of the Milstein scheme, which may be of independentinterest.

The Milstein scheme for singular SDEs with H\"older continuous drift