  The $d$-dimensional fractional Brownian motion (FBM for short)
$B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $\alpha$,
$\alpha\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process
with covariance ${\mathbb{E}}[B_s^{(i)}B
_t^{(j)}]={1/2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2 \alpha}).$ The
long-standing problem of defining a stochastic integration with respect to FBM
(and the related problem of solving stochastic differential equations driven by
FBM) has been addressed successfully by several different methods, although in
each case with a restriction on the range of either $d$ or $\alpha$. The case
$\alpha={1/2}$ corresponds to the usual stochastic integration with respect to
Brownian motion, while most computations become singular when $\alpha$ gets
under various threshhold values, due to the growing irregularity of the
trajectories as $\alpha\to0$. We provide here a new method valid for any $d$
and for $\alpha>{1/4}$ by constructing an approximation
$\Gamma(\varepsilon)_t$, $\varepsilon\to0$, of FBM which allows to define
iterated integrals, and then applying the geometric rough path theory. The
approximation relies on the definition of an analytic process $\Gamma_z$ on the
cut plane $z\in\mathbb{C}\setminus\mathbb{R}$ of which FBM appears to be a
boundary value, and allows to understand very precisely the well-known (see
\citeCQ02) but as yet a little mysterious divergence of L\'evy's area for
$\alpha\to{1/4}$.
