In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process on , that starts at zero, has expectation zero for all in , and has the following covariance function:

where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).

The value of H determines what kind of process the fBm is:

The increment process, X(t) = BH(t+1) − BH(t), is known as fractional Gaussian noise.

There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm.[1] n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm.

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.

Background and definition

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 to applications of fractional Brownian motion because of its over-emphasis of the origin (Mandelbrot & van Ness 1968, p. 424).

The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral

for t > 0 (and similarly for t < 0).

The main difference between fractional Brownian motion and regular Brownian motion is that while the increments in Brownian Motion are independent, increments for fractional Brownian motion are not. If H > 1/2, then there is positive autocorrelation: if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well. If H < 1/2, the autocorrelation is negative.

Properties

Self-similarity

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.

Stationary increments

It has stationary increments:

Long-range dependence

For H > ½ the process exhibits long-range dependence,

Regularity

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.

Dimension

With probability 1, the graph of BH(t) has both Hausdorff dimension[2] and box dimension[citation needed] of 2−H.

Integration

As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales.

Frequency-domain interpretation

Just as Brownian motion can be viewed as white noise filtered by (i.e. integrated), fractional Brownian motion is white noise filtered by (corresponding to fractional integration).

Sample paths

Practical computer realisations of an fBm can be generated,[3] although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an fBm process. Three realizations are shown below, each with 1000 points of an fBm with Hurst parameter 0.75.

"H" = 0.75 realisation 1
"H" = 0.75 realisation 2
"H" = 0.75 realisation 3

Realizations of three different types of fBm are shown below, each showing 1000 points, the first with Hurst parameter 0.15, the second with Hurst parameter 0.55, and the third with Hurst parameter 0.95. The higher the Hurst parameter is, the smoother the curve will be.

"H" = 0.15
"H" = 0.55
"H" = 0.95

Method 1 of simulation

One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies on the Cholesky decomposition method of the covariance matrix (explained below), which on a grid of size has complexity of order . A more complex, but computationally faster method is the circulant embedding method of Dietrich & Newsam (1997).

Suppose we want to simulate the values of the fBM at times using the Cholesky decomposition method.

In order to compute , we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of :

Note that the result is real-valued because .

Note that since the eigenvectors are linearly independent, the matrix is invertible.

Method 2 of simulation

It is also known that [4]

where B is a standard Brownian motion and

Where is the Euler hypergeometric integral.

Say we want to simulate an fBm at points .

The integral may be efficiently computed by Gaussian quadrature.

See also

Notes

  1. ^ Perrin et al., 2001.
  2. ^ Orey, 1970.
  3. ^ Kroese, D.P.; Botev, Z.I. (2014). "Spatial Process Generation". Lectures on Stochastic Geometry, Spatial Statistics and Random Fields, Volume II: Analysis, Modeling and Simulation of Complex Structures, Springer-Verlag, Berlin. arXiv:1308.0399. Bibcode:2013arXiv1308.0399K.
  4. ^ Decreusefond, L.; Üstünel, A. S. (1999). "Stochastic analysis of the fractional Brownian motion". CiteSeerX 10.1.1.201.5698. ((cite journal)): Cite journal requires |journal= (help)

References

Further reading