Section: Scientific Foundations
Fractional Brownian Motion
The Fractional Brownian Motion BH( t) with Hurst parameter Hhas originally been introduced by Kolmogorov for the study of turbulence. Since then many other applications have been found. If then BH( t) coincides with the standard Brownian motion, which has independent increments. If then BH( t) has a long memory or strong aftereffect . On the other hand, if , then BH( t) is anti-persistent : positive increments are usually followed by negative ones and vice versa. The strong aftereffect is often observed in the logarithmic returns for financial quantities Yn while the anti-persistence appears in turbulence and in the behavior of volatilities in finance.
For all H(0, 1) the process BH( t) is self-similar , in the sense that BH( t) has the same law as HBH( t) , for all >0 . Nevertheless, if , BH( t) is not a semi-martingale nor a Markov process  ,  ,  , and integration with respect to a FBM requires a specific stochastic integration theory.