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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 processwith covariance ${\mathbb{E}}[B_s^{(i)}B_t^{(j)}]={1/2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2 \alpha}).$ Thelong-standing problem of defining a stochastic integration with respect to FBM(and the related problem of solving stochastic differential equations driven byFBM) has been addressed successfully by several different methods, although ineach 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 toBrownian motion, while most computations become singular when $\alpha$ getsunder various threshhold values, due to the growing irregularity of thetrajectories 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 defineiterated integrals, and then applying the geometric rough path theory. Theapproximation relies on the definition of an analytic process $\Gamma_z$ on thecut plane $z\in\mathbb{C}\setminus\mathbb{R}$ of which FBM appears to be aboundary 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$\alpha\to{1/4}$.

Stochastic calculus for fractional Brownian motion with Hurst exponent H>¼: A rough path method by analytic extension