Fractional Brownian Motion

Abstract

We consider the integral of fractional Brownian motion (IFBM) and its functionals ξT on the intervals (0, T ) and (−T, T ) of the following types: the maximum MT , the position of the maximum, the occupation time above zero etc. We show how the asymptotics of P (ξT < 1) = pT , T → ∞, 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 pT . 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.