Fractional Brownian Motion_An Introdction with Necessary Mathmatics Tools: Malliavin Caculus and Wick Product

Fractional Brownian Motion

A normalized fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization ofBrownian motion without independent increments. It is a continuous-time Gaussian process BH(t) on [0, T], which starts at zero, has expectation zero for all t in [0, T], and has the following covariance function:

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

Prior to the introduction of the fractional Brownian motion,Lévy (1953) 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.The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral