Deep Neural Networks for the Fractional Fokker-PlanckEquation: Application to the Heston Model with FractionalBrownian Motion

Deep Neural Networks for the Fractional Fokker-PlanckEquation: Application to the Heston Model with FractionalBrownian Motion | 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 Deep Neural Networks for the Fractional Fokker-PlanckEquation: Application to the Heston Model with FractionalBrownian Motion Muhammed Ahmed Ibrahim, Tamirat Temesgen Dufera This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7710183/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 This paper investigates the application of a deep neural network approach for solving the fractionalFokker-Planck equation for the Heston model with fractional Brownian motion. In the framework of theHeston model, we first extend the Fokker-Planck equation with standard Brownian motion to a fractionalBrownian motion. Using state-of-the-art deep learning techniques, we design and train a deep neuralnetwork model and perform extensive tests. To increase the accuracy of the model, we compare the differentoptimization techniques, and Adams is more accurate for our problem in terms of losses and smoothness. Acomparison of activation functions reveals that Swish is the most effective choice to balance accuracy andcomputational efficiency. Our findings also highlight the flexibility of deep neural network-based Fokker-Planck equation models, showing that solution accuracy improves with different time points. Additionally,for various Hurst parameters 0 < H < 1, we demonstrate the robustness, scalability, and adaptabilityof the model. Furthermore, the deep neural network approach outperforms the finite difference method interms of error reduction and convergence. The results show that the solutions of both methods yield thebest agreement. In general, our results indicate that this method is effective, computationally efficient, andprovides comprehensive and standard solutions for stochastic volatility models in financial mathematics. Deep neural network Fractional Brownian motion Fokker-Planck equation Probability density function Fractional Fokker-Planck equation 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. 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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-7710183","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":527141506,"identity":"da306c54-a82b-4581-8fcd-2d67c50dee80","order_by":0,"name":"Muhammed Ahmed Ibrahim","email":"","orcid":"","institution":"Adama Science and Technology University","correspondingAuthor":false,"prefix":"","firstName":"Muhammed","middleName":"Ahmed","lastName":"Ibrahim","suffix":""},{"id":527141508,"identity":"029e266c-2c90-432f-9fe7-00bc279dd43c","order_by":1,"name":"Tamirat Temesgen Dufera","email":"data:image/png;base64,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","orcid":"","institution":"Adama Science and Technology University","correspondingAuthor":true,"prefix":"","firstName":"Tamirat","middleName":"Temesgen","lastName":"Dufera","suffix":""}],"badges":[],"createdAt":"2025-09-25 07:53:30","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7710183/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7710183/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93263580,"identity":"5f6d21cc-f124-49fc-a0f3-7ed8bb456c6e","added_by":"auto","created_at":"2025-10-10 19:02:33","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4234,"visible":true,"origin":"","legend":"","description":"","filename":"fbae10abc6614ba3bbca283a0821f807.json","url":"https://assets-eu.researchsquare.com/files/rs-7710183/v1/418ba2e9d58076035a0cb48e.json"},{"id":99314555,"identity":"cfb118df-53da-40fe-a805-b023027fb8dc","added_by":"auto","created_at":"2025-12-31 16:21:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2027243,"visible":true,"origin":"","legend":"","description":"","filename":"DNNFFPEWithFBM.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7710183/v1_covered_91c54680-b865-4347-87ca-196a76e25710.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Deep Neural Networks for the Fractional Fokker-PlanckEquation: Application to the Heston Model with FractionalBrownian Motion","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":"Deep neural network, Fractional Brownian motion, Fokker-Planck equation, Probability density function, Fractional Fokker-Planck equation","lastPublishedDoi":"10.21203/rs.3.rs-7710183/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7710183/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper investigates the application of a deep neural network approach for solving the fractionalFokker-Planck equation for the Heston model with fractional Brownian motion. In the framework of theHeston model, we first extend the Fokker-Planck equation with standard Brownian motion to a fractionalBrownian motion. Using state-of-the-art deep learning techniques, we design and train a deep neuralnetwork model and perform extensive tests. To increase the accuracy of the model, we compare the differentoptimization techniques, and Adams is more accurate for our problem in terms of losses and smoothness. Acomparison of activation functions reveals that Swish is the most effective choice to balance accuracy andcomputational efficiency. Our findings also highlight the flexibility of deep neural network-based Fokker-Planck equation models, showing that solution accuracy improves with different time points. Additionally,for various Hurst parameters 0 \u003c H \u003c 1, we demonstrate the robustness, scalability, and adaptabilityof the model. Furthermore, the deep neural network approach outperforms the finite difference method interms of error reduction and convergence. The results show that the solutions of both methods yield thebest agreement. 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