We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. https://doi.org/10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is \(H\in (0,1)\) and the drift is \(\mathcal {C}^\alpha \), \(\alpha \in [0,1]\) and \(\alpha >1-1/(2H)\), we show the strong \(L_p\) and almost sure rates of convergence to be \(((1/2+\alpha H)\wedge 1) -\varepsilon \), for any \(\varepsilon >0\). Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. https://doi.org/10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence \(1/2-\varepsilon \) of the Euler–Maruyama scheme for \(\mathcal {C}^\alpha \) drift, for any \(\varepsilon ,\alpha >0\).

我们对随机微分方程 (SDE) 的近似误差分析给出了新的看法，利用和发展了 Lê 的随机缝合引理（Electron J Probab 25:55, 2020. https://doi.org/10.1214/20- EJP442）。这种方法允许人们利用噪声效应的正则化来获得收敛速度。在我们的第一个应用程序中，我们展示了由具有非规则漂移的分数布朗运动驱动的 SDE 的 Euler-Maruyama 方案的收敛性（据我们所知是第一次）。当 Hurst 参数为\(H\in (0,1)\)且漂移为\(\mathcal {C}^\alpha \)、\(\alpha \in [0,1]\)和\( \alpha >1-1/(2H)\)，我们表明强\(L_p\)和几乎可以肯定的收敛速度为\(((1/2+\alpha H)\wedge 1) -\varepsilon \)，对于任何\(\varepsilon >0\)。我们关于漂移规律的条件是最佳的，因为它们与 Catellier 和 Gubinelli 的解的强唯一性所需的条件一致（Stoch Process Appl 126(8):2323–2366, 2016. https://doi .org/10.1016/j.spa.2016.02.002）。在第二个应用中，我们考虑由乘法标准布朗噪声驱动的 SDE 的近似值，其中我们推导出Euler-Maruyama 方案的几乎最佳收敛率\(1/2-\varepsilon \) \(\mathcal {C}^ \alpha \)漂移，对于任何\(\varepsilon ,\alpha >0\)。