%0 Journal Article
%A Rydin Gorjão, Leonardo
%A Witthaut, Dirk
%A Lehnertz, Klaus
%A Lind, Pedro G.
%T Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator
%J Entropy
%V 23
%N 5
%@ 1099-4300
%C Basel
%I MDPI
%M FZJ-2021-01852
%P 517 -
%D 2021
%X 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.
%F PUB:(DE-HGF)16
%9 Journal Article
%$ 33923154
%U <Go to ISI:>//WOS:000653859500001
%R 10.3390/e23050517
%U https://juser.fz-juelich.de/record/891975