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Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach

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  • Medino, Ary V.
  • Lopes, Sílvia R.C.
  • Morgado, Rafael
  • Dorea, Chang C.Y.

Abstract

Lévy processes have been widely used to model a large variety of stochastic processes under anomalous diffusion. In this note we show that Lévy processes play an important role in the study of the Generalized Langevin Equation (GLE). The solution to the GLE is proposed using stochastic integration in the sense of convergence in probability. Properties of the solution processes are obtained and numerical methods for stochastic integration are developed and applied to examples. Time series methods are applied to obtain estimation formulas for parameters related to the solution process. A Monte Carlo simulation study shows the estimation of the memory function parameter. We also estimate the stability index parameter when the noise is a Lévy process.

Suggested Citation

  • Medino, Ary V. & Lopes, Sílvia R.C. & Morgado, Rafael & Dorea, Chang C.Y., 2012. "Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 572-581.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:3:p:572-581
    DOI: 10.1016/j.physa.2011.09.025
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    References listed on IDEAS

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    1. Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2003. "Autocorrelation as a source of truncated Lévy flights in foreign exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 323(C), pages 601-625.
    2. Press, S. J., 1972. "Multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 2(4), pages 444-462, December.
    3. D. Brockmann & L. Hufnagel & T. Geisel, 2006. "The scaling laws of human travel," Nature, Nature, vol. 439(7075), pages 462-465, January.
    4. Dorea, Chang C.Y. & Guevara Otiniano, Cira E. & Matsushita, Raul & Rathie, Pushpa N., 2007. "Levy flight approximations for scaled transformations of random walks," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6343-6354, August.
    5. Luis Valdivieso & Wim Schoutens & Francis Tuerlinckx, 2009. "Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 1-19, February.
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    Cited by:

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    4. Karling, Maicon J. & Lopes, Sílvia R.C. & de Souza, Roberto M., 2023. "Multivariate α-stable distributions: VAR(1) processes, measures of dependence and their estimations," Journal of Multivariate Analysis, Elsevier, vol. 195(C).

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