We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order |$\kappa \in (0,1)$| for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order |$\min \{3/4,(1+\kappa )/2\}-\epsilon$| for the equidistant EM scheme (for arbitrarily small |$\epsilon>0$|⁠). The cut-off of the convergence order at |$3/4$| can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of |$(1+\kappa )/2-\epsilon$| for the corresponding EM scheme.

中文翻译：

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具有不规则漂移的SDE的Euler-Maruyama方案：通过简化为正交问题来收敛速度

对于具有加性噪声和不规则漂移的标量随机微分方程，我们研究了Euler-Maruyama（EM）方案的强收敛阶。通过将其简化为布朗运动的不规则函数的加权正交问题，我们为误差分析提供了一个通用框架。假设Sobolev–Slobodeckij型订单的正则性| $ \ kappa \ in（0,1）$ | 对于漂移的非平滑部分，我们对正交问题的分析得出收敛阶| $ \ min \ {3/4，（1+ \ kappa）/ 2 \}-\ epsilon $ | 等距EM方案（对于| $ \ epsilon> 0 $ |⁠任意小）。收敛阶的截止| $ 3/4 $ |可以通过使用适当的非等距离散化来克服，该离散化产生| $（1+ \ k）/ 2- \ epsilon $ |的强收敛阶。用于相应的EM方案。