We study strong approximation of scalar additive noise driven stochastic differential equations (SDEs) at time point $1$ in the case that the drift coefficient is bounded and has Sobolev regularity $s\in(0,1)$. Recently, it has been shown in [arXiv:2101.12185v2 (2022)] that for such SDEs the equidistant Euler approximation achieves an $L^2$-error rate of at least $(1+s)/2$, up to an arbitrary small $\varepsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In the present article we prove a matching lower error bound for $s\in(1/2,1)$. More precisely we show that, for every $s\in(1/2,1)$, the $L^2$-error rate $(1+s)/2$ can, up to a logarithmic term, not be improved in general by no numerical method based on finitely many evaluations of $W$ at fixed time points. Up to now, this result was known in the literature only for the cases $s=1/2-$ and $s=1-$. For the proof we employ the coupling of noise technique recently introduced in [arXiv:2010.00915 (2020)] to bound the $L^2$-error of an arbitrary approximation from below by the $L^2$-distance of two occupation time functionals provided by a specifically chosen drift coefficient with Sobolev regularity $s$ and two solutions of the corresponding SDE with coupled driving Brownian motions. For the analysis of the latter distance we employ a transformation of the original SDE to overcome the problem of correlated increments of the difference of the two coupled solutions, occupation time estimates to cope with the lack of regularity of the chosen drift coefficient around the point $0$ and scaling properties of the drift coefficient.

翻译：本文研究在时间点$1$处标量加性噪声驱动的随机微分方程（SDEs）的强逼近问题，其中漂移系数有界且具有Sobolev正则性$s\in(0,1)$。近期，文献[arXiv:2101.12185v2 (2022)]表明，对于此类SDEs，基于驱动布朗运动$W$的评估次数，等距欧拉逼近的$L^2$误差率至少达到$(1+s)/2$（可忽略任意小$\varepsilon$）。本文针对$s\in(1/2,1)$的情况证明匹配的下界。具体而言，我们证明：对于每个$s\in(1/2,1)$，基于在固定时间点上对$W$进行有限次评估的任何数值方法，均无法普遍改进$L^2$误差率$(1+s)/2$（对数量级项除外）。此前，文献中仅对$s=1/2-$和$s=1-$的情况已知此结果。在证明中，我们采用近期文献[arXiv:2010.00915 (2020)]引入的噪声耦合技术，通过特定选择具有Sobolev正则性$s$的漂移系数及对应SDE的两个解（耦合驱动布朗运动），将任意逼近的$L^2$误差下界归约为两个占位时泛函的$L^2$距离。为分析该距离，我们通过对原始SDE进行变换以克服两个耦合解差分的相关增量问题，利用占位时估计处理所选漂移系数在点$0$附近缺乏正则性的难题，并借助漂移系数的缩放性质。