Description: Arbitrary-Order Finite-Time Corrections for the Kramers

This title appears in the Scientific Report : 2021

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

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the differ...

Personal Name(s):	Rydin Gorjão, Leonardo (Corresponding author)
	Witthaut, Dirk / Lehnertz, Klaus / Lind, Pedro G.
Contributing Institute:	Systemforschung und Technologische Entwicklung; IEK-STE
Published in:	Entropy, 23 (2021) 5, S. 517 -
Imprint:	Basel MDPI 2021
DOI:	10.3390/e23050517
Document Type:	Journal Article
Research Program:	Energiesystemtransformation
Link:	Get full text OpenAccess
	Publikationsportal JuSER

Please use the identifier: http://hdl.handle.net/2128/27686 in citations.
Please use the identifier: http://dx.doi.org/10.3390/e23050517 in citations.

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.