We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

翻译：本文综述了基于驱动布朗运动的有限次评估，在最终时刻对随机微分方程解的p阶均值逐点逼近的复杂度领域最新进展。首先，我们简要回顾全局Lipschitz连续系数方程的情形——对于此类方程，使用等距Euler-Maruyama格式总能保证误差率至少达到驱动布朗运动评估次数的1/2次方。接着我们阐明：放弃系数的全局Lipschitz连续性可能导致Euler-Maruyama格式的误差呈非多项式衰减，甚至使得基于驱动布朗运动有限次评估所能实现的最小可能误差出现任意慢的衰减。最后，我们转向漂移系数非全局Lipschitz连续方程的相关积极结果。此处重点研究扩散系数Lipschitz连续、漂移系数满足分段光滑假设或具有分数阶Sobolev正则性的标量方程，并给出相应的复杂度结果。