  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
$H\rightarrow\frac{1}{2}$ 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 $H\rightarrow\frac{1}{2}$. 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
$\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if
$\{X_t^n,0\le t\le T\}$ 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
$H\in(\frac{1}{2},\frac{3}{4}]$. 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.
