Fractal (fractional) Brownian motion

Fractal Brownian motion, also called fractional Brownian motion (fBm), is a class of stochastic processes characterized by a single parameter called the Hurst parameter, which is a real number between zero and one. fBm becomes ordinary standard Brownian motion when the parameter has the value of onehalf. In this manner, it generalizes ordinary standard Brownian motion. Here, we precisely define fBm, compare it with Brownian motion, and describe its unique mathematical and statistical properties, including fractal behavior. Ideas of how such properties make these stochastic processes useful models of natural or manmade systems in life are described. We show how to use these processes as unique random noise representations in state equation models of some systems. We finally present statistical state equation estimation techniques where such processes replace traditional Gaussian white noises. (C) 2011 John Wiley & Sons, Inc.