Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $Hrightarrowfrac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It^{o} SDEs for $H=frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $Hrightarrowfrac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=frac{1}{2}$, and it has the rate of convergence $gamma_n^{-1}$, where $gamma_n=n^{2H-{1}/2}$ when $H<frac{3}{4}$, $gamma_n=n/sqrt{log n}$ when $H=frac{3}{4}$ and $gamma_n=n$ if $H>frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if ${X_t,0le tle T}$ is the solution of a SDE driven by a fBm and if ${X_t^n,0le tle T}$ is its approximation obtained by the new modified Euler scheme, then we prove that $gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $Hin(frac{1}{2},frac{3}{4}]$. In the case $H>frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.