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The class of nonlinear stochastic models as a background for the bursty behavior in financial markets

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  • Vygintas Gontis
  • Aleksejus Kononovicius
  • Stefan Reimann

Abstract

We investigate large changes, bursts, of the continuous stochastic signals, when the exponent of multiplicativity is higher than one. Earlier we have proposed a general nonlinear stochastic model which can be transformed into Bessel process with known first hitting (first passage) time statistics. Using these results we derive PDF of burst duration for the proposed model. We confirm analytical expressions by numerical evaluation and discuss bursty behavior of return in financial markets in the framework of modeling by nonlinear SDE.

Suggested Citation

  • Vygintas Gontis & Aleksejus Kononovicius & Stefan Reimann, 2012. "The class of nonlinear stochastic models as a background for the bursty behavior in financial markets," Papers 1201.3083, arXiv.org, revised May 2012.
    • Handle: RePEc:arx:papers:1201.3083
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    1. Chan, K C, et al, 1992. "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, vol. 47(3), pages 1209-1227, July.
    2. Vygintas Gontis & Bronislovas Kaulakys, 2003. "Multiplicative point process as a model of trading activity," Papers cond-mat/0303089, arXiv.org, revised Dec 2004.
    3. Alan Kirman, 1993. "Ants, Rationality, and Recruitment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(1), pages 137-156.
    4. 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.
    5. Reimann, St. & Gontis, V. & Alaburda, M., 2011. "Interplay between positive feedbacks in the generalized CEV process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(8), pages 1393-1401.
    6. Alfarano, Simone & Lux, Thomas & Wagner, Friedrich, 2008. "Time variation of higher moments in a financial market with heterogeneous agents: An analytical approach," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 101-136, January.
    7. Dmitry Davydov & Vadim Linetsky, 2001. "Pricing and Hedging Path-Dependent Options Under the CEV Process," Management Science, INFORMS, vol. 47(7), pages 949-965, July.
    8. Gontis, V. & Kaulakys, B., 2004. "Multiplicative point process as a model of trading activity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 505-514.
    9. Kaulakys, Bronislovas & Ruseckas, Julius & Gontis, Vygintas & Alaburda, Miglius, 2006. "Nonlinear stochastic models of 1/f noise and power-law distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 217-221.
    10. 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.
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    Citations

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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. Aleksejus Kononovicius & Vygintas Gontis, 2014. "Herding interactions as an opportunity to prevent extreme events in financial markets," Papers 1409.8024, arXiv.org, revised May 2015.
    3. Gontis, V. & Kononovicius, A., 2020. "Bessel-like birth–death process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    4. 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.
    5. Kononovicius, A. & Ruseckas, J., 2015. "Nonlinear GARCH model and 1/f noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 74-81.
    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. 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.
    8. Aleksejus Kononovicius & Vygintas Gontis, 2019. "Approximation of the first passage time distribution for the birth-death processes," Papers 1902.00924, arXiv.org.
    9. Aleksejus Kononovicius & Julius Ruseckas, 2014. "Nonlinear GARCH model and 1/f noise," Papers 1412.6244, arXiv.org, revised Feb 2015.
    10. 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.
    11. Vygintas Gontis & Aleksejus Kononovicius, 2019. "Bessel-like birth-death process," Papers 1904.13064, arXiv.org, revised Oct 2019.
    12. Vygintas Gontis & Aleksejus Kononovicius, 2017. "The consentaneous model of the financial markets exhibiting spurious nature of long-range memory," Papers 1712.05121, arXiv.org, revised Feb 2018.
    13. 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.
    14. Vygintas Gontis & Aleksejus Kononovicius, 2017. "Spurious memory in non-equilibrium stochastic models of imitative behavior," Papers 1707.09801, arXiv.org.
    15. V. Gontis & A. Kononovicius, 2017. "Burst and inter-burst duration statistics as empirical test of long-range memory in the financial markets," Papers 1701.01255, arXiv.org.

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