Fractional Brownian Motion

Fractional Brownian Motion

In probability theory, a normalized fractional Brownian motion (fBm), also called a fractal Brownian motion, is a Brownian motion without independent increments. It is a continuous-time Gaussian process B_H(t) on, which starts at zero, has expectation zero for all t 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:

• 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) = B_H(t+1) − B_H(t), is known as fractional Gaussian noise.

Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown, and fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.