Fractional Brownian Motion

In the nineties, statistical evidence, notably in finance and telecommunications, showed that Markov processes were too far away from the observations to be considered as viable models. In particular, there were strong suspicions that the data exhibit long range dependence. It is in this context that the fractional Brownian motion, introduced by B. Mandelbrot in the late sixties and almost forgotten since, enjoyed a new rise of interest. It is a Gaussian process with long range dependence. Consequently, it cannot be a semi-martingale, and we cannot apply the theory of Itô calculus. As we have seen earlier, for the Brownian motion, the Malliavin divergence generalizes the Itô integral and can be constructed for the fBm, so it is tempting to view it as an ersatz of a stochastic integral. Actually, the situation is not that simple and depends on what we call a stochastic integral.