Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation

[2607.27588] Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation Skip to main content Search Submit Donate Log in Quantitative Finance > Mathematical Finance arXiv:2607.27588 (q-fin) [Submitted on 30 Jul 2026 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation Authors: Damiano Brigo , Vladimir Lucic View PDF HTML (experimental) Abstract: This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker--Planck formulation that connects naturally with Rao--Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form. Subjects: Mathematical Finance (q-fin.MF) ; Probability (math.PR) MSC classes: 91G80, 60H15, 35R60 Cite as: arXiv:2607.27588 [q-fin.MF] (or arXiv:2607.27588v2 [q-fin.MF] for this version) https://doi.org/10.48550/arXiv.2607.27588 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Damiano Brigo [ view email ] [v1] Thu, 30 Jul 2026 02:16:32 UTC (23 KB) [v2] Mon, 24 Aug 2026 22:34:36 UTC (24 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: q-fin.MF < prev | next > new | recent | 2026-07 Change to browse by: math math.PR q-fin References & Citations SciX 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)

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