Fractional Brownian motion

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 ${\textstyle B_{H}(t)}$ on ${\textstyle [0,T]}$, that starts at zero, has expectation zero for all ${\displaystyle t}$ in ${\textstyle [0,T]}$, and has the following covariance function:

${\displaystyle E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),}$

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:

• 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.

Fractional Brownian motion has stationary increments X(t) = B_H(s+t) − B_H(s) (the value is the same for any s). The increment process X(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.