The Milstein scheme for singular SDEs with Hölder continuous drift

We study the $L^p$ rate of convergence of the Milstein scheme for SDEs when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularisation by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.