Fractional interacting particle system: drift parameter estimation via Malliavin calculus

Fractional interacting particle system: drift parameter estimation via Malliavin calculus | 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 Fractional interacting particle system: drift parameter estimation via Malliavin calculus Chiara Amorino, Ivan Nourdin, Radomyra Shevchenko This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6707122/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \( H \geq 1/2 \). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \( N \to \infty \). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \( H \in (0,1) \). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators. AMS 2010 subject classifications: Primary 62M09; secondary 60G18, 60H07, 60H10. Fractional Brownian motion interacting particle system Malliavin calculus drift parameter estimation McKean-Vlasov equations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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-6707122","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":469674769,"identity":"5cc362d2-8cab-4381-b24d-5f18785d63c8","order_by":0,"name":"Chiara Amorino","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVRIiWNgGAWjYDACCcYGOPsAA4MNAwMzkOIhQUsaAwNbAiEtqNzDhLXwz25u/FxRwyCv29778HBBxfnE7WwMjA/etuGx5M7BZskzxxgMt505bnB4xpnbiTvbGJgN5+LRYiCR2CDZADR42400hsO8bbcTN9xvYJPmxa+l+WfDPwZ7qJZziRuOMbD/JqClTbKxjSERquUASAsbMz4tEjcS2ywb+ySStwE9dJjnTLLxzjbGZsk553Br4Z+R/vhmwzcb223H25g/81TYyW5nYz744U0Zbi0wy5CcyoAUucQBAxLVj4JRMApGwfAHAMGFVIaIyuwHAAAAAElFTkSuQmCC","orcid":"","institution":"Pompeu Fabra University","correspondingAuthor":true,"prefix":"","firstName":"Chiara","middleName":"","lastName":"Amorino","suffix":""},{"id":469674770,"identity":"0cc230c6-d423-4907-a057-07261ec0095e","order_by":1,"name":"Ivan Nourdin","email":"","orcid":"","institution":"University of Luxembourg","correspondingAuthor":false,"prefix":"","firstName":"Ivan","middleName":"","lastName":"Nourdin","suffix":""},{"id":469674771,"identity":"aaf95578-e5a9-4ead-9c5d-908a8bedb2d2","order_by":2,"name":"Radomyra Shevchenko","email":"","orcid":"","institution":"Université Côte d'Azur","correspondingAuthor":false,"prefix":"","firstName":"Radomyra","middleName":"","lastName":"Shevchenko","suffix":""}],"badges":[],"createdAt":"2025-05-20 11:08:25","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6707122/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6707122/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":90217689,"identity":"21343f13-1f94-4ea7-ba6e-80060b5bf2c2","added_by":"auto","created_at":"2025-08-30 07:16:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":771367,"visible":true,"origin":"","legend":"","description":"","filename":"main.tex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6707122/v1_covered_ba7144d5-f0c0-404c-b305-11f94004b999.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Fractional interacting particle system: drift parameter estimation via Malliavin calculus","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fractional Brownian motion, interacting particle system, Malliavin calculus, drift parameter estimation, McKean-Vlasov equations","lastPublishedDoi":"10.21203/rs.3.rs-6707122/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6707122/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \\( H \\geq 1/2 \\). 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