# Definition:

Given a complete probability space  $\left(\Omega, \mathcal{F}, \mathbb{P} \right)$.

A fractional Brownian motion (fBm)  $\left( B_t^H, t \geq 0\right)$  of Hurts parameter  $H \in (0,1)$ is the continuous centered Gaussian process with covariance function

$R^{H}(t,s):=\mathbf{E}\left[ B_t^{H} B_s^{H} \right] =\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$. (1)

The parameter $H$characterizes all the important properties of the process.

Note that  $E\left[(B_t^H)^2 \right]=|t|^{2H}$ and then $\mathbf{E}\left[ (B_t^H- B_s^H)^2\right]= |t-s|^{2H}$.  As $\left( B^H(t), t \geq 0 \right)$ is Gaussian so he admits a version with  continuous trajectories or continuous modification, according to the Kolmogorov theorem.

For $H=1$, we set $B_t^H= B_t^1= t \xi$ where $\xi$ is standard normal random variable.

The  parameter  $H$ controls the regularity of the trajectories, which are Hölder continuous of order $H- \varepsilon$ , for any $\varepsilon >0$. more precisely,

For all $\varepsilon > 0$ and $\alpha > 0$, there exists a nonnegative random variable  $X_{\varepsilon, \alpha}$ such that $E\left[ |X_{\varepsilon, \alpha}|^p\right] < \infty$  for all   $p \geq 1$, and

$|B_t^H -B_s^H| \leq X_{\varepsilon, \alpha} |t-s|^{H - \varepsilon}$

This is simply the  modulus of continuity for the trajectories of a  fBm $\left( B_t^H, t \geq 0 \right)$ .

If $H=\frac{1}{2}$, the covariance  $R_{\frac{1}{2}}(t,s)= \min(t,s)$  and the process $\left(B_t^H , t \geq 0\right)$ is a standard brownian motion.  in this case, the increments of this process  in disjoint intervals are independents.

### Self-similarity

An $\mathbb{R}^d$-valued random  process $X= (X_t, t \in \mathbb{R})$ is  $b$self-similar  if  for any  $a >0$ there exists    $b > 0$  such that

$\displaystyle\left(X_{a t}, t \in \mathbb{R} \right)$  and  $\displaystyle\left( a^bX_{ t}, t \in \mathbb{R} \right)$  have the same distribution.

that means for, for every choise $t_0, \cdots , t_n$ in  $\mathbb{R}$  we have,

$\displaystyle{\mathbb{P}\left( X_{a t_0} \leq x_0, \cdots , X_{a t_n} \leq x_n\right) = \mathbb{P \left( bX_{ t_0} \leq x_0, \cdots , b X_{t_n} \leq x_n\right)}$

for every $x_0, \cdots , x_n$ in  $\mathbb{R}$.

Since the covariance function of the fBm is homogenous of order $2 H$, we  deduce that the process $\left( B_t^{H}, t \geq 0 \right)$ is self-similar (Put $b=a^{-H}$).

Note that

$\mathbf{E}\left[(B_t^{H} -B_s^{H} )(B_u^{H} -B_v^{H} )\right]= \frac{1}{2}\left[|s-u|^{2 H} + |t -v|^{2 H} - |t -u|^{2 H} - |s -v|^{2 H}\right]$   (2)

and it follow that the process  $\left( B_t^{H} , t \geq 0 \right)$ has stationary increment,  However it  is not stationary itself.

Let $H \in ]0, \frac{1}{2}[ \cup ]\frac{1}{2}, 1[$ and $t_1 < t_2 < t_3 < t_4$, it follow from  (2)  that

$\mathbf{E}\left[(B_{t_4}^{H} - B_{t_3}^{H} )(B_{t_2}^{H} - B_{t_1}^{H} ) \right]= H(2 H -1) \int_{t_1}^{t_2} \, \int_{t_3}^{t_4} (u-v)^{2 H-2} \, du \, dv$.