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Arbitrary-Order Finite-Time Corrections for the Kramers-Moyal Operator.

Leonardo Rydin Gorjão1,2, Dirk Witthaut1,2, Klaus Lehnertz3,4,5

  • 1Forschungszentrum Jülich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, Germany.

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Summary

This study introduces a new method to reconstruct stochastic differential equations from time-series data. It improves accuracy by including finite-time corrections for diffusion and jump-diffusion processes.

Keywords:
Bell polynomialsFokker–Planck equationKramers–Moyal coefficientsKramers–Moyal equationarbitrary-order approximationsnon-parametric estimatorsstochastic processes

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Area of Science:

  • Stochastic processes
  • Time-series analysis
  • Mathematical physics

Background:

  • Reconstructing stochastic evolution equations from empirical time-series data is challenging.
  • Existing methods often struggle with finite sampling intervals and discontinuous processes.

Purpose of the Study:

  • To improve the reconstruction of stochastic evolution equations from time-series data.
  • To develop a method that accounts for finite-time corrections and discontinuous processes.

Main Methods:

  • Derived a full representation of the Kramers-Moyal operator generator using a power-series expansion.
  • Separated finite-time corrections into terms with and without derivatives of Kramers-Moyal coefficients.
  • Developed a closed-form solution using conditional moments extractable from time-series data.

Main Results:

  • Provided all finite-time correction terms for parametric and non-parametric estimation of Kramers-Moyal coefficients.
  • Demonstrated the method's effectiveness for diffusion and jump-diffusion processes, even with insufficient sampling.
  • Showcased the ability to distinguish between diffusion and jump-diffusion processes using only time-series data.

Conclusions:

  • The proposed method offers arbitrary-order finite-time corrections for enhanced accuracy in stochastic process reconstruction.
  • The approach, utilizing Bell polynomials, is readily implementable in time-series analyses.
  • This work provides a robust framework for analyzing complex stochastic systems from empirical data.