On the Entropy of Fractionally Integrated Gauss-Markov Processes

This paper is devoted to the estimation of the entropy of the dynamical system {X-alpha(t),t >= 0}, where the stochastic process X-alpha(t) consists of the fractional Riemann-Liouville integral of order alpha is an element of(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 alpha and the complexity degree; we show that the entropy of X-alpha(t) is a decreasing function of alpha is an element of(0,1).