Strong rate of convergence for the Euler-Maruyama scheme of additive fractional SDEs with Lipschitz drift

Abstract

We study the strong convergence rate of the Euler-Maruyama scheme for additive stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in (0,1)$. Assuming the drift coefficient to be Lipschitz continuous, we show that the rate is $1$ if $H\in (1/2,1)$, and $1/2+H-\epsilon$, for any $\epsilon >0$, if $H\in (0,1/2]$. The main ingredient is a shifted stochastic sewing argument, which exploits the conditional Gaussian structure of fractional Brownian motion to control the noise discretization error.