The Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise

Abstract

In this paper we consider the Euler-Maruyama scheme for a class ofstochastic delay differential equations driven by a fractional Brownian motion with index $H\in(0,1)$. We establish the consistency of the scheme and study the rate of convergence of the normalized error process. This is done by checking that the generic rate of convergence of the error process with stepsize $\Delta_{n}$ is $\Delta_{n}^{\min\{H+\frac{1}{2},3H,1\}}$. It turned out that such a rate is suboptimal when the delay is smooth and $H>1/2$. In this context, and in contrast to the non-delayed framework, we show that a convergence of order $H+1/2$ is achievable.