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On the Entropy of Fractionally Integrated Gauss–Markov Processes

Author

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  • Mario Abundo

    (Dipartimento di Matematica, Università “Tor Vergata”, via della Ricerca Scientifica, I-00133 Roma, Italy
    These authors contributed equally to this work.)

  • Enrica Pirozzi

    (Dipartimento di Matematica e Applicazioni, Università “Federico II”, via Cintia, Complesso Monte S. Angelo, I-80126 Napoli, Italy
    These authors contributed equally to this work.)

Abstract

This paper is devoted to the estimation of the entropy of the dynamical system { X α ( t ) , t ≥ 0 } , where the stochastic process X α ( 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 X α ( t ) is a decreasing function of α ∈ ( 0 , 1 ) .

Suggested Citation

  • Mario Abundo & Enrica Pirozzi, 2020. "On the Entropy of Fractionally Integrated Gauss–Markov Processes," Mathematics, MDPI, vol. 8(11), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2031-:d:445044
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    References listed on IDEAS

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    1. Abundo, Mario, 2012. "An inverse first-passage problem for one-dimensional diffusions with random starting point," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 7-14.
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