## Section: Scientific Foundations

### Fractional Brownian Motion

The Fractional Brownian Motion
B_{H}(
t) with Hurst parameter
Hhas 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
[86] ,
[67] ,
[68] , and integration with respect to a FBM requires a specific stochastic integration theory.