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Section: Scientific Foundations

Fractional Brownian Motion

Participant : A. Sulem.

The Fractional Brownian Motion BH(t) with Hurst parameter H has 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 [68] , [52] , [53] , and integration with respect to a FBM requires a specific stochastic integration theory.


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