On the entropy of fractionally integrated Gauss–Markov processes

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1)

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

Non-Local Solvable Birth-Death Processes

In this paper we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes.Comment: 26 page

Simulation of sample paths for Gauss-Markov processes in the presence of a reflecting boundary

Algorithms for the simulation of sample paths of Gauss–Markov processes, restricted from below by particular time-dependent reflecting boundaries, are proposed. These algorithms are used to build the histograms of first passage time density through specified boundaries and for the estimation of related moments. Particular attention is dedicated to restricted Wiener and Ornstein–Uhlenbeck processes due to their central role in the class of Gauss–Markov processes