Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.