Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

Abstract

In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of $\hat{I}»$-Hölder continuous functions on $[-\ddot{I}„,0]$, for some suitable $\ddot{I}„>0$ and $\hat{I}»\in (1/2,H)$. The rate of convergence of our scheme is $1/n^3$, for any ${\hat{I}}^3<2\hat{I}»-1$. Also, numerical simulations are provided to illustrate our theoretical results.

30 pages, 1 figure