Given a complete probability space .
A fractional Brownian motion (fBm) of Hurts parameter is the continuous centered Gaussian process with covariance function
The parameter characterizes all the important properties of the process.
Note that and then . As is Gaussian so he admits a version with continuous trajectories or continuous modification, according to the Kolmogorov theorem.
For , we set where is standard normal random variable.
The parameter controls the regularity of the trajectories, which are Hölder continuous of order , for any . more precisely,
For all and , there exists a nonnegative random variable such that for all , and
This is simply the modulus of continuity for the trajectories of a fBm .
If , the covariance and the process is a standard brownian motion. in this case, the increments of this process in disjoint intervals are independents.
An -valued random process is -self-similar if for any there exists such that
and have the same distribution.
that means for, for every choise in we have,
for every in .
Since the covariance function of the fBm is homogenous of order , we deduce that the process is self-similar (Put ).
and it follow that the process has stationary increment, However it is not stationary itself.
Let and , it follow from (2) that