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：我们研究了在时间点$1$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$L^2$误差率至少为$(1+s)/2$（允许任意小的$\varepsilon$）。在本文中，我们针对$s\in(1/2,1)$的情况证明了匹配的下界。具体而言，我们证明：对于每个$s\in(1/2,1)$，任何基于固定时间点有限次评估$W$的数值方法通常无法将$L^2$误差率$(1+s)/2$提升至超越对数项的水平。此前，文献中仅已知$s=1/2-$和$s=1-$两种情况的结果。在证明中，我们采用文献[arXiv:2010.00915 (2020)]中最新引入的噪声耦合技术，通过下界任意逼近的$L^2$误差与两个占用时间泛函的$L^2$距离之间的关系（该距离由特定选取的具有Sobolev正则性$s$的漂移系数及对应SDE的两个解（由耦合驱动布朗运动生成）确定）。为分析后者距离，我们采用原SDE的变换以克服两个耦合解差分的相关增量问题，利用占用时间估计处理所选漂移系数在点$0$附近缺乏正则性的困难，并借助漂移系数的缩放性质。