{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T03:59:35Z","timestamp":1778212775047,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,24]],"date-time":"2021-04-24T00:00:00Z","timestamp":1619222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["03EK3055B"],"award-info":[{"award-number":["03EK3055B"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001656","name":"Helmholtz-Gemeinschaft","doi-asserted-by":"publisher","award":["ZT-I-0029"],"award-info":[{"award-number":["ZT-I-0029"]}],"id":[{"id":"10.13039\/501100001656","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Moyal 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\u2013Moyal 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\u2013Moyal coefficients for discontinuous processes which can be easily implemented\u2014employing Bell polynomials\u2014in 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.<\/jats:p>","DOI":"10.3390\/e23050517","type":"journal-article","created":{"date-parts":[[2021,4,24]],"date-time":"2021-04-24T21:49:20Z","timestamp":1619300960000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Arbitrary-Order Finite-Time Corrections for the Kramers\u2013Moyal Operator"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5513-0580","authenticated-orcid":false,"given":"Leonardo","family":"Rydin Gorj\u00e3o","sequence":"first","affiliation":[{"name":"Forschungszentrum J\u00fclich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 J\u00fclich, Germany"},{"name":"Institute for Theoretical Physics, University of Cologne, 50937 K\u00f6ln, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3623-5341","authenticated-orcid":false,"given":"Dirk","family":"Witthaut","sequence":"additional","affiliation":[{"name":"Forschungszentrum J\u00fclich, Institute for Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52428 J\u00fclich, Germany"},{"name":"Institute for Theoretical Physics, University of Cologne, 50937 K\u00f6ln, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5529-8559","authenticated-orcid":false,"given":"Klaus","family":"Lehnertz","sequence":"additional","affiliation":[{"name":"Department of Epileptology, University Hospital Bonn, Venusberg Campus 1, 53127 Bonn, Germany"},{"name":"Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14\u201316, 53115 Bonn, Germany"},{"name":"Interdisciplinary Center for Complex Systems, University of Bonn, Br\u00fchler Stra\u00dfe 7, 53175 Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8176-666X","authenticated-orcid":false,"given":"Pedro G.","family":"Lind","sequence":"additional","affiliation":[{"name":"Department of Computer Science, OsloMet\u2014Oslo Metropolitan University, P.O. Box 4 St. Olavs plass, N-0130 Oslo, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/S0031-8914(40)90098-2","article-title":"Brownian motion in a field of force and the diffusion model of chemical reactions","volume":"7","author":"Kramers","year":"1940","journal-title":"Physica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1111\/j.2517-6161.1949.tb00030.x","article-title":"Stochastic processes and statistical physics","volume":"11","author":"Moyal","year":"1949","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"35435","DOI":"10.1038\/srep35435","article-title":"Disentangling the stochastic behavior of complex time series","volume":"6","author":"Anvari","year":"2016","journal-title":"Sci. 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