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

Abstract

We consider the integral of fractional Brownian motion (IFBM) and its functionals $ξ_T$ 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 $P(ξ_T<1)=p_T, T\to \infty$, 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 $θ$ of the power law is related to the similarity parameter $H$ of fractional Brownian motion as follows: $θ=-(1-H)$ for the interval $(-T,T)$ and $θ=-H(1-H)$ for $(0,T)$. The point 0 is special in that IFBM and its derivative both vanish there.