{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-formalization-of-martingales-in-isabelle","title":"A Formalization of Martingales in Isabelle/HOL","arxiv_id":"2311.06188","date":"2023-11-10","proceeding":null,"authors":["Ata Keskin"],"abstract":"This thesis presents a formalization of martingales in arbitrary Banach spaces using Isabelle/HOL. We begin by examining formalizations in prominent proof repositories and extend the definition of the conditional expectation operator from the real numbers to general Banach spaces. The current formalization of conditional expectation in the Isabelle library is limited to real-valued functions. To overcome this limitation, we use measure theoretic arguments to construct the conditional expectation in Banach spaces using suitable limits of simple functions. Subsequently, we define stochastic processes and introduce the concepts of adapted, progressively measurable and predictable processes using suitable locale definitions. We show the relation $$\\text{adapted} \\supseteq \\text{progressive} \\supseteq \\text{predictable}$$ Furthermore, we show that progressive measurability and adaptedness are equivalent when the indexing set is discrete. We pay special attention to predictable processes in discrete-time, showing that $(X_n)_{n \\in \\mathbb{N}}$ is predictable if and only if $(X_{n + 1})_{n \\in \\mathbb{N}}$ is adapted. We rigorously define martingales, submartingales, and supermartingales, presenting their first consequences and corollaries. Discrete-time martingales are given special attention in the formalization. In every step of our formalization, we make extensive use of the powerful locale system of Isabelle. The formalization further contributes by generalizing concepts in Bochner integration by extending their application from the real numbers to arbitrary Banach spaces equipped with a second-countable topology. Induction schemes for integrable simple functions on Banach spaces are introduced. Additionally, we formalize a powerful result called the \"Averaging Theorem\" which allows us to show that densities are unique in Banach spaces.","url_abs":"https://arxiv.org/abs/2311.06188v1","url_pdf":"https://arxiv.org/pdf/2311.06188v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-formalization-of-martingales-in-isabelle","repo_url":"https://github.com/atakeskinn/martingales","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}