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 alpha is an element of (0, 1], X-alpha(t), we investigate the decaying rate of P(tau(alpha)(S) > t) as t -> +infinity, where tau(alpha)(S) = inf{t > 0 : X-alpha(t) >= S} is the first-passage time (FPT) of X-alpha(t) through the barrier S > 0. Precisely, we study the so-called persistent exponent theta = theta(alpha) of the FPT tail, such that P(tau(alpha)(S) > t) = t(-theta+o(1)), as t -> +infinity, and by means of numerical simulation of long enough trajectories of the process X-alpha(t), we are able to estimate theta(alpha) and to show that it is a non-increasing function of alpha is an element of (0, 1], with 1/4 <= theta(alpha) <= 1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function theta = theta(alpha), for alpha is an element of (0, 1]. Such a numerical validation is carried out in two ways: in the first one, we estimate theta(alpha), by using the simulated FPT density, obtained for any alpha is an element of (0, 1]; in the second one, we estimate the persistent exponent by directly calculating P(max(0)<= s <= tX(alpha)(s) < 1). Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of X-alpha(t) and we find the upper bound of its covariance function.