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

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.