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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 Im5 ${H=\mfrac 12}$ then BH( t) coincides with the standard Brownian motion, which has independent increments. If Im6 ${H\gt \mfrac 12}$ then BH( t) has a long memory or strong aftereffect . On the other hand, if Im7 ${0\lt H\lt \mfrac 12}$ , 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 Im8 ${log\mfrac Y_nY_{n-1}}$ for financial quantities Yn while the anti-persistence appears in turbulence and in the behavior of volatilities in finance.

For all H$ \in$(0, 1) the process BH( t) is self-similar , in the sense that BH( $ \alpha$t) has the same law as $ \alpha$HBH( t) , for all $ \alpha$>0 . Nevertheless, if Im9 ${H\#8800 \mfrac 12}$ , BH( t) is not a semi-martingale nor a Markov process [86] , [67] , [68] , and integration with respect to a FBM requires a specific stochastic integration theory.


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