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The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem
IMA Journal of Numerical Analysis ( IF 2.3 ) Pub Date : 2020-06-17 , DOI: 10.1093/imanum/draa007
Andreas Neuenkirch 1 , Michaela Szölgyenyi 2
Affiliation  

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:

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。
更新日期:2020-06-17
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