Fractal Itô Calculus: Extensions and Applications

Fractal Itô Calculus: Extensions and Applications | 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 Fractal Itô Calculus: Extensions and Applications Alireza Khalili Golmankhaneh, Donatella Bongiorno, Carlo Cattani, and 4 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6804587/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract In this paper, we first summarize fractal calculus and extend It^{o} calculus to fractal sets, focusing on the integration and differentiation of stochastic processes within fractal structures. We compare Brownian motion on the real line with that on a ternary Cantor set, generalizing Ito's framework to accommodate fractal geometry complexities. We define the fractal Ito integral with respect to fractal Brownian motion and establish the fractal Ito lemma, forming a foundation for fractal stochastic differential equations. Applications span finance and physics, including modeling stock prices with a Fractal Black-Scholes equation and simulating particle movement and population dynamics on fractals. Itô Calculus on Fractals Fractal Brownian Motion Fractal Itô Integral Fractal Black-Scholes Model Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 06 Jun, 2025 Reviewers invited by journal 03 Jun, 2025 Editor assigned by journal 03 Jun, 2025 Submission checks completed at journal 03 Jun, 2025 First submitted to journal 02 Jun, 2025 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-6804587","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":466068569,"identity":"aafedf19-1a48-4955-86ea-87fd0e8d8467","order_by":0,"name":"Alireza Khalili Golmankhaneh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVRIie3RMWrDMBSA4ScEzqKS9YUY+wovdGiHklxFJYMXFzwWGlqDwV1CuybkFLmBwOAsPUDHhEKnDB49uFDJuIQOcjp20L9IID6ekABcrv/YgKXtighMyQQDrzsQVsJPBJQkvPwD6daWAMFteu5ew4znVdVMYbTJ9mpP19HL4I2gXoB/ZcFYsGy9EnMY+yWZi93lIia2LEH4yjJGEy6QQ4CyIxATXKQg0CJCQxp60iSqDIm84ZHYVw8hQ0AWMMa4nSI9veF9UyaasKXaidEqTgyZ5PiZFH6JVhLsnj+gbh4CfI+2h/r+MQxf59vDcXEzs5Gffn+c0k95BrhcLperr29yrkyfQ3pVeQAAAABJRU5ErkJggg==","orcid":"","institution":"Ur. 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