Fractional Brownian motion
Unlike classical Brownian motion, the increments of fBm need not be independent.
The value of H determines what kind of process the fBm is: Fractional Brownian motion has stationary increments X(t) = BH(s+t) − BH(s) (the value is the same for any s).
The increment process X(t) is known as fractional Gaussian noise.
[1] n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary.
Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.
Prior to the introduction of the fractional Brownian motion, Lévy (1953) used the Riemann–Liouville fractional integral to define the process where integration is with respect to the white noise measure dB(s).
This integral turns out to be ill-suited as a definition of fractional Brownian motion because of its over-emphasis of the origin (Mandelbrot & van Ness 1968, p. 424).
The process is self-similar, since in terms of probability distributions: This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property.
FBm can also be defined as the unique mean-zero Gaussian process, null at the origin, with stationary and self-similar increments.
It has stationary increments: For H > 1/2 the process exhibits long-range dependence, Sample-paths are almost nowhere differentiable.
However, almost-all trajectories are locally Hölder continuous of any order strictly less than H: for each such trajectory, for every T > 0 and for every ε > 0 there exists a (random) constant c such that for 0 < s,t < T. With probability 1, the graph of BH(t) has both Hausdorff dimension[2] and box dimension[3] of 2−H.
As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals".
Just as Brownian motion can be viewed as white noise filtered by
(i.e. integrated), fractional Brownian motion is white noise filtered by
Practical computer realisations of an fBm can be generated,[4][5] although they are only a finite approximation.
Three realizations are shown below, each with 1000 points of an fBm with Hurst parameter 0.75.
The higher the Hurst parameter is, the smoother the curve will be.
One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function.
Suppose we want to simulate the values of the fBM at times
Note that since the eigenvectors are linearly independent, the matrix
The integral may be efficiently computed by Gaussian quadrature.
