Team-Mathfi:Fractional Brownian Motion

Inria / Raweb 2006
Presentation of the Team Mathfi

Section: Scientific Foundations

Fractional Brownian Motion

Participant : A. Sulem.

The Fractional Brownian Motion B_H(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}$$ then B_H(t) coincides with the standard Brownian motion, which has independent increments. If $Im6 ${H\gt \mfrac 12}$$ then B_H(t) has a long memory or strong aftereffect . On the other hand, if $Im7 ${0\lt H\lt \mfrac 12}$$ , then B_H(t) is anti-persistent : positive increments are usually followed by negative ones and vice versa. The strong aftereffect is often observed in the logarithmic returns $Im8 ${log\mfrac Y_nY_{n-1}}$$ for financial quantities Y_n while the anti-persistence appears in turbulence and in the behavior of volatilities in finance.

For all H $\in$ (0, 1) the process B_H(t) is self-similar , in the sense that B_H( $\alpha$ t) has the same law as $\alpha$ ^HB_H(t) , for all $\alpha$ >0 . Nevertheless, if $Im9 ${H\#8800 \mfrac 12}$$ , B_H(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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