Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises

Abstract

The strong convergence rate of the Euler scheme for stochastic differential equations driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter H∈(13,12) and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the 2D 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+12 for linear cases.