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In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter 1312. The current contribution generalizes the modified Euler scheme to the rough case 13<H<12. Namely, we show a convergence rate of order n^1{2-2H} for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Hölder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
Liu et al. (Thu,) studied this question.