The strong convergence rate of the Euler scheme for SDEs driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the $2$D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order $2H$ for the case that the drift coefficient has bounded derivatives up to order three and have strong order $H+\frac12$ for linear cases. Numerical simulations are presented to support the theoretical results.
翻译:研究了由加分分数布朗动议驱动的Euler SDEs计划强烈的趋同率,其中分数布朗运动拥有赫斯特参数$H\in(frac13,\frac12)$和漂移系数,无需加以约束。马利亚温微积分、粗路径理论和$D Young 集成物被利用来克服分数布朗运动的低规律性和漂移系数的无约束性造成的困难。事实证明,Euler计划具有强烈的2H$,因为流系数将衍生物捆绑到3级,线性案例的强烈排序为$Hzfrac12$。提供了数字模拟来支持理论结果。