[2311.07936] Occupied Processes: Going with the Flow Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2311.07936 (math) [Submitted on 14 Nov 2023 ( v1 ), last revised 28 Apr 2026 (this version, v6)] Title: Occupied Processes: Going with the Flow Authors: Valentin Tissot-Daguette View a PDF of the paper titled Occupied Processes: Going with the Flow, by Valentin Tissot-Daguette View PDF HTML (experimental) Abstract: A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$ that tracks the time spent by the path at each level. When $X$ is Markov, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure as well. We develop an Itô calculus for occupied processes that lies midway between Dupire's functional Itô calculus and the classical version. We derive Itô formulae and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. The framework's benefits are illustrated via an optimal stopping problem involving local times, followed by financial applications. For the latter, we show how occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price derivatives books with a single numerical solver. We finally explore an extension of forward variance models so as to leverage the entire forward occupation surface. Comments: 45 pages Subjects: Probability (math.PR) ; Mathematical Finance (q-fin.MF); Pricing of Securities (q-fin.PR) MSC classes: 60J55, 60J25, 60H30, 60G40, 91G20 Cite as: arXiv:2311.07936 [math.PR] (or arXiv:2311.07936v6 [math.PR] for this version) https://doi.org/10.48550/arXiv.2311.07936 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Stochastic Processes and their Applications, 195:104890, 2026 Related DOI : https://doi.org/10.1016/j.spa.2026.104890 Focus to learn more DOI(s) linking to related resources Submission history From: Valentin Tissot-Daguette [ view email ] [v1] Tue, 14 Nov 2023 06:27:06 UTC (2,809 KB) [v2] Fri, 8 Dec 2023 02:44:10 UTC (2,809 KB) [v3] Mon, 25 Aug 2025 23:23:47 UTC (6,593 KB) [v4] Fri, 29 Aug 2025 17:29:41 UTC (6,593 KB) [v5] Sat, 11 Oct 2025 20:31:01 UTC (6,744 KB) [v6] Tue, 28 Apr 2026 19:55:51 UTC (2,956 KB) Full-text links: Access Paper: View a PDF of the paper titled Occupied Processes: Going with the Flow, by Valentin Tissot-Daguette View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2023-11 Change to browse by: math q-fin q-fin.MF q-fin.PR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from