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

For a fractionally integrated Brownian motion (FIBM) of order (Formula presented.) (Formula presented.) we investigate the decaying rate of (Formula presented.) as (Formula presented.) where (Formula presented.) is the first-passage time (FPT) of (Formula presented.) through the barrier (Formula presented.) Precisely, we study the so-called persistent exponent (Formula presented.) of the FPT tail, such that (Formula presented.) as (Formula presented.) and by means of numerical simulation of long enough trajectories of the process (Formula presented.) we are able to estimate (Formula presented.) and to show that it is a non-increasing function of (Formula presented.) with (Formula presented.) In particular, we are able to validate numerically a new conjecture about the analytical expression of the function (Formula presented.) for (Formula presented.) Such a numerical validation is carried out in two ways: in the first one, we estimate (Formula presented.) by using the simulated FPT density, obtained for any (Formula presented.) in the second one, we estimate the persistent exponent by directly calculating (Formula presented.) Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of (Formula presented.) and we find the upper bound of its covariance function.