The Financial Bubble Model with Lévy Jump Processes

[2606.25104] The Financial Bubble Model with Lévy Jump Processes Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2606.25104 (math) [Submitted on 23 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: The Financial Bubble Model with Lévy Jump Processes Authors: Avetik Arakelyan , Rafayel Barkhudaryan , Vigen Khalatyan , Michael Poghosyan View PDF HTML (experimental) Abstract: In this work we consider an extension of the Berestycki--Monneau--Scheinkman (BMS) model for speculative financial bubbles, in which the investor disagreement process is allowed to have jumps. While the original BMS framework assumes that disagreement evolves along continuous paths, our model accounts for sudden shifts in market sentiment through an independent Lévy jump process. Using optimal stopping theory and the Itô--Lévy formula, we show that the speculative bubble premium satisfies a non-local partial integro-differential equation (PIDE) with a moving obstacle. We develop a viscosity solution theory for this non-local obstacle problem. We prove a comparison principle by a doubling-of-variables argument adapted to the non-local jump integral, and we prove existence and uniqueness of the bubble price by Perron's method, constructing explicit continuous sub- and supersolutions. We then introduce a monotone Implicit--Explicit finite difference scheme for the bubble premium. Following the Barles--Souganidis framework, we show that the discrete operator preserves the M-matrix property and that the scheme converges locally uniformly to the unique viscosity solution under a state-dependent Courant--Friedrichs--Lewy (CFL) condition. At the end of the paper we implement the scheme via a PSOR--Picard algorithm and present numerical tests for four finite- and infinite-activity Lévy models. Comments: 35 pages, 6 Figures, Keywords: Financial Bubbles, Levy Process, Viscosity Solutions, Free Boundary Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2606.25104 [math.AP] (or arXiv:2606.25104v3 [math.AP] for this version) https://doi.org/10.48550/arXiv.2606.25104 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Avetik Arakelyan Ara [ view email ] [v1] Tue, 23 Jun 2026 19:20:14 UTC (933 KB) [v2] Tue, 14 Jul 2026 08:13:52 UTC (920 KB) [v3] Thu, 27 Aug 2026 10:09:35 UTC (920 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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

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