On the Entropy of Fractionally Integrated Gauss–Markov Processes

Abstract

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).

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“…And the integral t 0 e β s dW s can also be represented as Weiner process 1 √ 2β W e 2βt − 1 (for more details, see reference 23 ).…”

“…(8) Substituting the definition of u t = dx t dt , and integrating with respect to time yields 12,23 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.…”

“…where u t is the particle velocity, β = γ/m, |µ| = |2g(ρ f − ρ p )a 2 /(9µ)| is the Stokes velocity of a particle with density ρ p in a fluid with density ρ f , σ = 2γk B T /m, and W t is the Wiener process 19,23 .…”