On the Asymptotic Moment Preservation of the Stochastic Theta Method for Multi-dimensional Ornstein-Uhlenbeck Processes

On the Asymptotic Moment Preservation of the Stochastic Theta Method for Multi-dimensional Ornstein-Uhlenbeck Processes | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On the Asymptotic Moment Preservation of the Stochastic Theta Method for Multi-dimensional Ornstein-Uhlenbeck Processes Min Li, Kaiyan Fu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8786623/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract The preservation of long-term statistical properties is a key criterion for evaluating numerical method of stochastic differential equations. This paper gives the detailed and self-contained analysis of the stochastic theta method (STM) applied to multi-dimensional Ornstein-Uhlenbeck processes. Extending moment-preservation results from the one-dimensional to the multi-dimensional setting presents a major challenge, as it requires a shift from scalar analysis to an operator-theoretic framework based on continuous and discrete algebraic Lyapunov equations. Our main contribution lies in bridging this gap. We demonstrate that the STM preserves the asymptotic mean if and only if the method is mean-stable. This stability condition holds unconditionally for the implicit range of the parameter $\theta \in [1/2, 1]$, while a step-size restriction is required for parameter $\theta \in [0, 1/2)$. More importantly, by deriving the precise structure of the global error’s leading term, we prove that the trapezoidal scheme ($\theta=1/2$) is the only method in the STM family that achieves second-order accuracy for the asymptotic covariance matrix, while all other methods (STM with $\theta \neq 1/2$) exhibit only first-order accuracy.Finally, a two-dimensional colored noise model is included to verify our theoretical findings. Multi-dimensional Ornstein-Uhlenbeck process Stochastic theta method Algebraic Lyapunov equation Long term behavior Moment preservation. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 20 Feb, 2026 Editor assigned by journal 11 Feb, 2026 Submission checks completed at journal 10 Feb, 2026 First submitted to journal 04 Feb, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8786623","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":594401103,"identity":"1590a9d4-89de-4405-8c9e-f50a2cdb2930","order_by":0,"name":"Min Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYBACPmYGhgMghgED84EDH34QoYUNqkXCgIEt8eDMHmK0QGmgFh7jwxxseBVDtbBzJx4uzLGrM+c/8+EwAw+DPL/YAUIO491weOa2ZAnLGbkbDhdYMBjOnJ1AhBbebcwSBjeAjBk8DAkGt4nTUi9hcP7Mg8M8bMRrOSxhcCCHgSQtxyU33EgzAAayBGG/8POf3fyZd1s1v8H5w48/fPhhI88vTUALOpAgTfkoGAWjYBSMAuwAAOC3Qli3FKmfAAAAAElFTkSuQmCC","orcid":"","institution":"China University of Geosciences","correspondingAuthor":true,"prefix":"","firstName":"Min","middleName":"","lastName":"Li","suffix":""},{"id":594401105,"identity":"5439dae1-a3d8-4912-9d07-9d10ff2bb998","order_by":1,"name":"Kaiyan Fu","email":"","orcid":"","institution":"Shenzhen University","correspondingAuthor":false,"prefix":"","firstName":"Kaiyan","middleName":"","lastName":"Fu","suffix":""}],"badges":[],"createdAt":"2026-02-04 12:40:01","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8786623/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8786623/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104546651,"identity":"da20c39f-494c-4d87-adf6-4d6d828e0a1a","added_by":"auto","created_at":"2026-03-13 07:27:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":523626,"visible":true,"origin":"","legend":"","description":"","filename":"thetamOUmomentpreserve2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8786623/v1_covered_f7f3215e-05f1-4869-8026-bba936b544ed.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On the Asymptotic Moment Preservation of the Stochastic Theta Method for Multi-dimensional Ornstein-Uhlenbeck Processes","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Multi-dimensional Ornstein-Uhlenbeck process, Stochastic theta method, Algebraic Lyapunov equation, Long term behavior, Moment preservation.","lastPublishedDoi":"10.21203/rs.3.rs-8786623/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8786623/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The preservation of long-term statistical properties is a key criterion for evaluating numerical method of stochastic differential equations. 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