Deterministic versus Stochastic Inventory Modeling: A Brownian Motion–Based Stochastic Differential Equation Approach

Deterministic versus Stochastic Inventory Modeling: A Brownian Motion–Based Stochastic Differential Equation Approach | 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 Deterministic versus Stochastic Inventory Modeling: A Brownian Motion–Based Stochastic Differential Equation Approach Sourav Nath, Sumit Saha This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8015515/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 study develops and analyzes a stochastic differential equation (SDE)--based inventory model that extends traditional deterministic formulations to account for natural demand variability represented by Brownian motion. The objective is to examine how random fluctuations influence total cost dynamics and optimal replenishment decisions. The deterministic model is first formulated for a stock-dependent demand rate and then extended to a stochastic framework by incorporating an Itô process for random demand perturbations. Using Itô's lemma and stochastic integration, the analytical expressions for expected inventory level and Average Total Cost (ATC) are derived. Monte Carlo simulation is employed to numerically estimate the stochastic cost trajectories and validate the model's stability under uncertainty. Comparative results reveal that both deterministic and stochastic models yield an identical optimal cycle length ( \((T^{*} = 37.54)\) months), whereas the stochastic model achieves a lower expected ATC by approximately 2.2%, demonstrating enhanced cost efficiency and robustness. Sensitivity analysis confirms that the optimal decisions are stable under moderate variations in demand volatility. The findings are consistent with recent stochastic inventory studies based on the Hamilton--Jacobi--Bellman framework, validating the proposed model's reliability. Overall, the research establishes that integrating Brownian motion into inventory dynamics provides a realistic representation of demand uncertainty and supports more resilient cost optimization in uncertain operating environments. Inventory Management Stochastic Differential Equations (SDE) Brownian Motion Deterministic vs. Stochastic Models Demand Uncertainty Supply Chain Optimization. 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. 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Approach","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":"Inventory Management, Stochastic Differential Equations (SDE), Brownian Motion, Deterministic vs. Stochastic Models, Demand Uncertainty, Supply Chain Optimization.","lastPublishedDoi":"10.21203/rs.3.rs-8015515/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8015515/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study develops and analyzes a stochastic differential equation (SDE)--based inventory model that extends traditional deterministic formulations to account for natural demand variability represented by Brownian motion. The objective is to examine how random fluctuations influence total cost dynamics and optimal replenishment decisions. The deterministic model is first formulated for a stock-dependent demand rate and then extended to a stochastic framework by incorporating an It\u0026ocirc; process for random demand perturbations. Using It\u0026ocirc;'s lemma and stochastic integration, the analytical expressions for expected inventory level and Average Total Cost (ATC) are derived. Monte Carlo simulation is employed to numerically estimate the stochastic cost trajectories and validate the model's stability under uncertainty. Comparative results reveal that both deterministic and stochastic models yield an identical optimal cycle length (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((T^{*} = 37.54)\\)\u003c/span\u003e\u003c/span\u003e months), whereas the stochastic model achieves a lower expected ATC by approximately 2.2%, demonstrating enhanced cost efficiency and robustness. Sensitivity analysis confirms that the optimal decisions are stable under moderate variations in demand volatility. The findings are consistent with recent stochastic inventory studies based on the Hamilton--Jacobi--Bellman framework, validating the proposed model's reliability. Overall, the research establishes that integrating Brownian motion into inventory dynamics provides a realistic representation of demand uncertainty and supports more resilient cost optimization in uncertain operating environments.\u003c/p\u003e","manuscriptTitle":"Deterministic versus Stochastic Inventory Modeling: A Brownian Motion–Based Stochastic Differential Equation Approach","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-20 17:45:25","doi":"10.21203/rs.3.rs-8015515/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"966df5ac-94cb-4fda-9b9f-e9c203beac83","owner":[],"postedDate":"November 20th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-11T11:08:56+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-20 17:45:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8015515","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8015515","identity":"rs-8015515","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}