We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). 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, Gubinelli (2016). 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 $\varepsilon,\alpha>0$.
翻译:使用并开发L ⁇ e (202020年) 的缝纫利玛(L ⁇ e (202020年)) 。 这种方法允许一种人利用噪音效应进行正规化, 获得趋同率。 在第一个应用程序中, 我们展示了(我们第一次了解的)由不定期漂移的棕色分数动作驱动的Euler- Maruyama SDE计划趋同率。 当赫斯特参数是H$( 0. 1) 美元, 漂移值是$/ mathal=Calpha$( 0, 1美元) 和 ALpha> 1-1/ (2H) 美元时, 我们展示了强大的L_ p$和几乎肯定的趋同率是$( 1/2 ) -\\\ weng) 1 -\ varepsilon$, 任何非常规漂移值美元。 我们的规律性条件是最佳的, 因为它们与卡塔利西亚 $( ALphal) $ (Guban) IMillion) 的解决方案的强烈独特性需要的条件相匹配。 在第二个我们考虑Sal- Exliversalal- Exlimal 的 的 10 标准 上, 。