On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations

Abstract

For a fractionally integrated Brownian motion (FIBM) of order α∈(0,1],Xα(t), we investigate the decaying rate of P(τSα>t) as t→+∞, where τSα=inf{t>0:Xα(t)≥S} is the first-passage time (FPT) of Xα(t) through the barrier S>0. Precisely, we study the so-called persistent exponent θ=θ(α) of the FPT tail, such that P(τSα>t)=t−θ+o(1), as t→+∞, and by means of numerical simulation of long enough trajectories of the process Xα(t), we are able to estimate θ(α) and to show that it is a non-increasing function of α∈(0,1], with 1/4≤θ(α)≤1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function θ=θ(α), for α∈(0,1]. Such a numerical validation is carried out in two ways: in the first one, we estimate θ(α), by using the simulated FPT density, obtained for any α∈(0,1]; in the second one, we estimate the persistent exponent by directly calculating Pmax0≤s≤tXα(s)<1. Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of Xα(t) and we find the upper bound of its covariance function.