In recent years, interest in approximation methods for stochastic differential equations (SDEs) with non-Lipschitz continuous coefficients has increased. We show lower bounds for the $L^p$-error of such methods in the case of approximation at a single point in time or globally in time. On the one hand, we show that for a large class of piecewise Lipschitz continuous drifts and non-additive diffusions the best possible $L^p$-error rate for final time approximation that can be achieved by any method based on finitely many evaluations of the driving Brownian motion is at most $3/4$, which was previously known only for additive diffusions. Moreover, we show that the best $L^p$-error rate for global approximation that can be achieved by any method based on finitely many evaluations of the driving Brownian motion is at most $1/2$ when the drift is locally bounded and the diffusion is locally Lipschitz continuous. For the derivation of the lower bounds we introduce a new method of proof: the local coupling of noise technique. Using this technique when approximating a solution $X$ of the SDE at the final time, a lower bound for the $L^p$-error of any approximation method based on evaluations of the driving Brownian motion at the points $t_1 < \dots < t_n$ can be determined by the $L^p$-distances of solutions of the same SDE on $[t_{i-1}, t_i]$ with initial values $X_{t_{i-1}}$ and driving Brownian motions that are coupled at $t_{i-1}, t_i$ and independent, conditioned on the values of the Brownian motion at $t_{i-1}, t_i$.

翻译：近年来，对于具有非Lipschitz连续系数的随机微分方程（SDEs）逼近方法的研究兴趣日益增长。本文针对单时间点逼近和全局时间逼近两种情况，证明了此类方法$L^p$误差的下界。一方面，我们证明对于一大类分段Lipschitz连续漂移项和非加性扩散项，任何基于驱动布朗运动有限次评估的逼近方法所能达到的终时逼近最佳可能$L^p$误差阶至多为$3/4$——该结论此前仅在加性扩散情形下已知。此外，当漂移项局部有界且扩散项局部Lipschitz连续时，我们证明任何基于驱动布朗运动有限次评估的逼近方法所能达到的全局逼近最佳$L^p$误差阶至多为$1/2$。为推导这些下界，我们引入了一种新的证明方法：局部噪声耦合技术。该技术在逼近SDE解$X$的终值时，任何基于驱动布朗运动在点$t_1 < \dots < t_n$处取值的逼近方法，其$L^p$误差下界可通过计算以下解的$L^p$距离确定：这些解属于同一SDE在区间$[t_{i-1}, t_i]$上的解，其初始值分别为$X_{t_{i-1}}$，且对应的驱动布朗运动在$t_{i-1}$和$t_i$处耦合，并在给定$t_{i-1}$和$t_i$处布朗运动取值的条件下相互独立。