## Section: Scientific Foundations

### Fractional Brownian Motion

Participant : A. Sulem.

The Fractional Brownian Motion B_{H}(t) with Hurst parameter H has
originally been introduced by Kolmogorov for the study of turbulence.
Since then many other applications have been found.
If then B_{H}(t) coincides with the standard
Brownian motion, which has independent increments. If then B_{H}(t) has a *long memory* or *strong
aftereffect* . On the other hand, if , then
B_{H}(t) is *anti-persistent* : positive increments are usually followed by negative ones and vice versa.
The strong aftereffect is often observed in the logarithmic returns
for financial quantities Y_{n} while the
anti-persistence appears in turbulence and in the behavior of
volatilities in finance.

For all H(0, 1) the process B_{H}(t) is *self-similar* , in
the sense that B_{H}(t) has the same law as ^{H}B_{H}(t) ,
for all >0 .
Nevertheless, if , B_{H}(t) is not a semi-martingale
nor a Markov process [68] , [52] , [53] , and
integration with respect to a FBM requires a specific stochastic
integration theory.