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Bessel-like birth–death process

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  • Gontis, V.
  • Kononovicius, A.

Abstract

We consider models of the population or opinion dynamics which result in the non-linear stochastic differential equations (SDEs) exhibiting the spurious long-range memory. In this context, the correspondence between the description of the birth–death processes as the continuous-time Markov chains and the continuous SDEs is of high importance for the alternatives of modeling. We propose and generalize the Bessel-like birth–death process having clear representation by the SDEs. The new process helps to integrate the alternatives of description and to derive the equations for the probability density function (PDF) of the burst and inter-burst duration of the proposed continuous time birth–death processes.

Suggested Citation

  • Gontis, V. & Kononovicius, A., 2020. "Bessel-like birth–death process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    • Handle: RePEc:eee:phsmap:v:540:y:2020:i:c:s0378437119317595
    DOI: 10.1016/j.physa.2019.123119
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    References listed on IDEAS

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    1. Gontis, V. & Havlin, S. & Kononovicius, A. & Podobnik, B. & Stanley, H.E., 2016. "Stochastic model of financial markets reproducing scaling and memory in volatility return intervals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1091-1102.
    2. Aleksejus Kononovicius & Julius Ruseckas, 2014. "Continuous transition from the extensive to the non-extensive statistics in an agent-based herding model," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(8), pages 1-7, August.
    3. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    4. Vygintas Gontis & Aleksejus Kononovicius, 2014. "Consentaneous Agent-Based and Stochastic Model of the Financial Markets," PLOS ONE, Public Library of Science, vol. 9(7), pages 1-12, July.
    5. Alan Kirman, 1993. "Ants, Rationality, and Recruitment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(1), pages 137-156.
    6. Gontis, V. & Kononovicius, A., 2017. "Burst and inter-burst duration statistics as empirical test of long-range memory in the financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 266-272.
    7. Kononovicius, A. & Gontis, V., 2012. "Agent based reasoning for the non-linear stochastic models of long-range memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1309-1314.
    8. Aleksejus Kononovicius & Vygintas Gontis, 2019. "Approximation of the first passage time distribution for the birth-death processes," Papers 1902.00924, arXiv.org.
    9. Vygintas Gontis & Aleksejus Kononovicius & Stefan Reimann, 2012. "The Class Of Nonlinear Stochastic Models As A Background For The Bursty Behavior In Financial Markets," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 15(supp0), pages 1-13.
    10. Gontis, V. & Kononovicius, A., 2018. "The consentaneous model of the financial markets exhibiting spurious nature of long-range memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 1075-1083.
    11. Simone Alfarano & Thomas Lux & Friedrich Wagner, 2005. "Estimation of Agent-Based Models: The Case of an Asymmetric Herding Model," Computational Economics, Springer;Society for Computational Economics, vol. 26(1), pages 19-49, August.
    12. Alfarano, Simone & Milakovic, Mishael, 2009. "Network structure and N-dependence in agent-based herding models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 78-92, January.
    13. Oualid Jouini & Yves Dallery, 2008. "Moments of first passage times in general birth–death processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 49-76, August.
    14. Marta C. González & César A. Hidalgo & Albert-László Barabási, 2009. "Understanding individual human mobility patterns," Nature, Nature, vol. 458(7235), pages 238-238, March.
    15. Julius Ruseckas & Vygintas Gontis & Bronislovas Kaulakys, 2012. "Nonextensive Statistical Mechanics Distributions And Dynamics Of Financial Observables From The Nonlinear Stochastic Differential Equations," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 15(supp0), pages 1-13.
    16. V. Gontis & B. Kaulakys, 2006. "Long-range memory model of trading activity and volatility," Papers physics/0606115, arXiv.org.
    17. Vygintas Gontis & Aleksejus Kononovicius, 2017. "Spurious memory in non-equilibrium stochastic models of imitative behavior," Papers 1707.09801, arXiv.org.
    18. Vygintas Gontis & Aleksejus Kononovicius & Stefan Reimann, 2012. "The Class Of Nonlinear Stochastic Models As A Background For The Bursty Behavior In Financial Markets," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 15(supp0), pages 1-13.
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    Cited by:

    1. Vygintas Gontis, 2021. "Order flow in the financial markets from the perspective of the Fractional L\'evy stable motion," Papers 2105.02057, arXiv.org, revised Nov 2021.
    2. Rytis Kazakeviv{c}ius & Aleksejus Kononovicius, 2023. "Anomalous diffusion and long-range memory in the scaled voter model," Papers 2301.08088, arXiv.org, revised Feb 2023.
    3. Rytis Kazakevicius & Aleksejus Kononovicius & Bronislovas Kaulakys & Vygintas Gontis, 2021. "Understanding the nature of the long-range memory phenomenon in socioeconomic systems," Papers 2108.02506, arXiv.org, revised Aug 2021.

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