Fractional Brownian Motion

Fractional Brownian motion is a generalization of ordinary Brownian motion in which the increments of the process are no longer required to be independent of one another. It is formally defined as a continuous-time Gaussian process that starts at zero, has an expected value of zero at every time, and has a covariance function built around a parameter H, called the Hurst parameter, which ranges between 0 and 1 and controls how rough or smooth the resulting path looks: higher values of H produce smoother trajectories. Setting H to exactly one half recovers standard Brownian motion, while values above one half produce positively correlated increments and values below one half produce negatively correlated ones. The process was introduced by Benoit Mandelbrot and John van Ness in 1968 and has stationary increments known as fractional Gaussian noise.