Fractional Brownian motion

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.

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