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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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    Cited by:

    1. Montagnon, C.E., 2021. "Forecasting by splitting a time series using Singular Value Decomposition then using both ARMA and a Fokker Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    2. 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).
    3. Montagnon, Chris, 2017. "Forecasting with the Fokker–Planck model: Bayesian setting of parameter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 253-262.
    4. Montagnon, Chris, 2015. "A closed solution to the Fokker–Planck equation applied to forecasting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 14-22.

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