标题： 噪声局部耦合技术及其在SDEs（随机微分方程）不规则系数强逼近的误差下界中的应用 显示英文标题

标题： The local coupling of noise technique and its application to lower error bounds for strong approximation of SDEs with irregular coefficients

摘要： 近年来，人们对带有非Lipschitz连续系数的随机微分方程（SDE）的逼近方法越来越感兴趣。 我们展示了在这种情况下，单个时间点或全局时间上的这些方法的$L^p$-误差的下界。 一方面，我们证明了对于一大类分段Lipschitz连续漂移和非加性扩散，基于有限次驱动布朗运动评估的任何方法所能达到的最终时间逼近的最佳可能$L^p$-误差率至多为$3/4$，这一结果之前仅对加性扩散已知。 此外，我们还表明，当漂移局部有界且扩散局部Lipschitz连续时，基于有限次驱动布朗运动评估的任何方法所能达到的全局逼近的最佳$L^p$-误差率至多为$1/2$。 为了推导出这些下界，我们引入了一种新的证明方法：局部耦合噪声技术。 利用这一技术在最终时间逼近随机微分方程 (SDE) 的解$X$时，可以通过相同 SDE 在$[t_{i-1}, t_i]$上具有初始值$X_{t_{i-1}}$和耦合于$t_{i-1}, t_i$并在$t_{i-1}, t_i$处条件独立的驱动布朗运动的解之间的$L^p$-距离，确定基于在点$t_1 < \dots < t_n$评估驱动布朗运动的任何逼近方法的$L^p$-误差下界。 显示英文摘要

摘要： 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 $t_{i-1}, t_i$.