We study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations 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 regularization 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.

中文翻译：

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具有 Hölder 连续漂移的奇异 SDE 的 Milstein 方案


我们研究了当漂移系数仅具有 Hölder 规律时，随机微分方程的 Milstein 方案的 $L^{p}$ 收敛率。如果扩散是椭圆的并且足够规则，则我们得到与加法情况一致的速率。该证明依赖于噪声技术的正则化，特别是随机缝纫，这反过来又需要（至少渐近地）对米尔斯坦方案定律的敏锐估计，这可能具有独立的兴趣。