Rough Measure Theory: A Pathwise Foundation for Stochastic Analysis
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| Format: | Recurso digital |
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Zenodo
2025
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| _version_ | 1866901731258925056 |
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper proposes the development of a novel mathematical framework, termed "Rough Measure Theory," designed to provide a robust pathwise foundation for stochastic analysis. Traditional Ito calculus, while foundational for processes like Brownian motion, faces significant limitations when dealing with signals that are not semimartingales, particularly those exhibiting "roughness" or finite p-variation for p > 2. Rough Path Theory, pioneered by Terry Lyons, has revolutionized the understanding and analysis of such systems by lifting a rough path to a "geometric rough path," enabling the well-defined integration of vector fields. However, a comprehensive measure-theoretic analogue that operates inherently on the space of rough paths, rather than relying on an underlying probability space, has remained elusive. Our work aims to bridge this gap by constructing a new measure-theoretic framework where sigma-algebras and measures are defined directly on spaces of rough paths. This allows for a pathwise formulation of key concepts, offering a deterministic yet robust approach to stochastic analysis. We detail the theoretical construction of "rough sigma-algebras" and "rough measures," demonstrating how a rough integral can be rigorously defined within this context. The proposed theory promises to extend the applicability of stochastic analysis to a broader class of processes and systems, including those arising from complex dynamical systems, singular SPDEs, and high-frequency financial data, by providing a foundation that is intrinsic to the path's geometry rather than its probabilistic generation. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17839655 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Rough Measure Theory: A Pathwise Foundation for Stochastic Analysis Revista, Zen MATH, 10 This paper proposes the development of a novel mathematical framework, termed "Rough Measure Theory," designed to provide a robust pathwise foundation for stochastic analysis. Traditional Ito calculus, while foundational for processes like Brownian motion, faces significant limitations when dealing with signals that are not semimartingales, particularly those exhibiting "roughness" or finite p-variation for p > 2. Rough Path Theory, pioneered by Terry Lyons, has revolutionized the understanding and analysis of such systems by lifting a rough path to a "geometric rough path," enabling the well-defined integration of vector fields. However, a comprehensive measure-theoretic analogue that operates inherently on the space of rough paths, rather than relying on an underlying probability space, has remained elusive. Our work aims to bridge this gap by constructing a new measure-theoretic framework where sigma-algebras and measures are defined directly on spaces of rough paths. This allows for a pathwise formulation of key concepts, offering a deterministic yet robust approach to stochastic analysis. We detail the theoretical construction of "rough sigma-algebras" and "rough measures," demonstrating how a rough integral can be rigorously defined within this context. The proposed theory promises to extend the applicability of stochastic analysis to a broader class of processes and systems, including those arising from complex dynamical systems, singular SPDEs, and high-frequency financial data, by providing a foundation that is intrinsic to the path's geometry rather than its probabilistic generation. |
| title | Rough Measure Theory: A Pathwise Foundation for Stochastic Analysis |
| url | https://doi.org/10.5281/zenodo.17839655 |