Adaptive Stochastic Gradient Flow: A Variational Framework for Optimal Continuous-Time Approximation of Stochastic Optimization

Adaptive Stochastic Gradient Flow: A Variational Framework for Optimal Continuous-Time Approximation of Stochastic Optimization | 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 Adaptive Stochastic Gradient Flow: A Variational Framework for Optimal Continuous-Time Approximation of Stochastic Optimization Diplav Dongre, Ujwal Warbhe, Nileshchandra Kalbarao Pikle This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7834030/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 introduces a novel high-order numerical methodology for solving the generalized Black-Scholes equation with variable coefficients. The proposed approach combines hp-adaptive spectral element discretization in space with implicit-explicit (IMEX) time integration schemes, addressing key challenges in financial derivative pricing. The spatial discretization employs an adaptive strategy that dynamically refines both mesh size (h) and polynomial degree (p) based on a posteriori error estimation, effectively resolving the singularity originating from the non-differentiable payoff function of European options. The temporal discretization utilizes third-order L-stable IMEX additive Runge-Kutta (IMEX-ARK) methods, which implicitly handle the stiff diffusion term while explicitly treating advection and reaction components. We establish comprehensive theoretical foundations including well-posedness analysis, stability proofs, and convergence estimates under appropriate regularity conditions. Numerical experiments demonstrate the method's efficacy for various financial scenarios including constant coefficient models, local volatility surfaces, and time-dependent parameters. The framework is extended to exotic options, particularly barrier options and American options via a penalty formulation. Comparative studies against established numerical techniques confirm the superior accuracy and computational efficiency of the proposed methodology across all test cases. MSC Classification: 62L20 , 65C30 , 68T05 , 90C15 Stochastic Gradient Descent Continuous-Time Approximation Stochastic Differential Equations Weak Approximation Variational Methods Statistical Learning 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-7834030","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":534210460,"identity":"9a4fda1a-37bd-4df1-8fa1-a5a827914170","order_by":0,"name":"Diplav Dongre","email":"","orcid":"","institution":"Indian Institute of Information Technology, Nagpur","correspondingAuthor":false,"prefix":"","firstName":"Diplav","middleName":"","lastName":"Dongre","suffix":""},{"id":534210461,"identity":"609320e7-caab-433c-bb7d-229cae1fde3b","order_by":1,"name":"Ujwal Warbhe","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYJACZsYGBjk2EAnhEqnFGKilsYEkLYlA5TBrCACDG8nPPhfusEnvY2duf1zAYCfPwM57gICWNOPZM8+k5bYBHdY8gyHZsIGZLwG/ltsJxsy8bYchWngYmBMYmHkMCGhJ/wzU8j+dDaKlnhgtOSBbDiRAtRwmrEXy/pti5pltyYYgh83mMTgOZBDQwnfm+GbmwjY7efn+4w8+81RUy/Pzn8GvBd2dDAxspKgfBaNgFIyCUYAdAADsFDjP1qXmIwAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Information Technology, Nagpur","correspondingAuthor":true,"prefix":"","firstName":"Ujwal","middleName":"","lastName":"Warbhe","suffix":""},{"id":534210462,"identity":"fffd7bc5-aa95-47af-985e-f5d5a6ca2ec0","order_by":2,"name":"Nileshchandra Kalbarao Pikle","email":"","orcid":"","institution":"Indian Institute of Information Technology, Nagpur","correspondingAuthor":false,"prefix":"","firstName":"Nileshchandra","middleName":"Kalbarao","lastName":"Pikle","suffix":""}],"badges":[],"createdAt":"2025-10-11 09:38:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7834030/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7834030/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":94579281,"identity":"f969a618-9dab-4641-9557-79adc5e5095d","added_by":"auto","created_at":"2025-10-28 18:11:17","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5147,"visible":true,"origin":"","legend":"","description":"","filename":"895f3e82d6c14a58874bae39387a556e.json","url":"https://assets-eu.researchsquare.com/files/rs-7834030/v1/e34c219d9bece5efe9e52087.json"},{"id":106401626,"identity":"d3c4b132-a1ef-475e-aeeb-28a5720293ff","added_by":"auto","created_at":"2026-04-08 09:08:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":662713,"visible":true,"origin":"","legend":"","description":"","filename":"SpringerNatureLaTeXTemplate131.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7834030/v1_covered_9e44a61f-28ca-4287-9db0-bc56b3dcb495.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Adaptive Stochastic Gradient Flow: A Variational Framework for Optimal Continuous-Time Approximation of Stochastic Optimization","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":"Stochastic Gradient Descent, Continuous-Time Approximation, Stochastic Differential Equations, Weak Approximation, Variational Methods, Statistical Learning","lastPublishedDoi":"10.21203/rs.3.rs-7834030/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7834030/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper introduces a novel high-order numerical methodology for solving the generalized Black-Scholes equation with variable coefficients. 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