Background and Definition
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 because of its over-emphasis of the origin (Mandelbrot & van Ness 1968, p. 424).
The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral
for t > 0 (and similarly for t < 0).
The main difference between fractional Brownian motion and regular Brownian motion is that while the increments in Brownian Motion are independent, the opposite is true for fractional Brownian motion. This dependence means that if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well.
Read more about this topic: Fractional Brownian Motion
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