Small values of the maximum for the integral of fractional Brownian motion

Details Small values of the maximum for the integral of fractional Brownian motion Small values of the maximum for the integral of fractional Brownian motion

2002-12-19

arxiv.org/abs/math/0212281v1

Mathematics

Probability

23 pages,3 figures, TeX/LaTeX 3.14159

Scientific paper

We consider the integral of fractional Brownian motion (IFBM) and its functionals $\xi_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following types: the maximum $M_T$, the position of the maximum, the occupation time above zero etc. We show how the asymptotics of $P(\xi_T<1)=p_T, T\to \infty$, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for $p_T$. The data do not reject the hypothesis that the exponent $\theta$ of the power law is related to the similarity parameter $H$ of fractional Brownian motion as follows: $\theta =-(1-H)$ for the interval $(-T,T)$ and $\theta =-H(1-H)$ for $(0,T)$. The point 0 is special in that IFBM and its derivative both vanish there.

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