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Concept
Fractional Brownian motion
Summary
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 .
The value of H determines what kind of process the fBm is:
if H = 1/2 then the process is in fact a Brownian motion or Wiener process;
if H > 1/2 then the increments of the process are positively correlated;
if H < 1/2 then the increments of the process are negatively correlated.
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. 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.
Prior to the introduction of the fractional Brownian motion, 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 .
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.
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