A Fubini type theorem for rough integration
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Thomas R. Cass [2] ; Jeffrey Pei [1]
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[1] Imperial College London
Imperial College London
Reino Unido
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[2] Imperial College London; and The Alan Turing Institute, London, UK
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 2, 2023, págs. 761-802
- Idioma: inglés
- DOI: 10.4171/RMI/1409
- Enlaces
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Resumen
Jointly controlled paths as used in Hairer and Gerasimovičs (2019), are a class of two-parameter paths Y controlled by a p-rough path X for 2≤p<3 in each time variable, and serve as a class of paths twice integrable with respect to X. We extend the notion of jointly controlled paths to two-parameter paths Y controlled by p-rough and ~p~-rough paths X and ~X~ (on finite dimensional spaces) for arbitrary p and ~p~, and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and ~X~, they are integrable with respect to X and ~X~, and moreover we prove a rough Fubini type theorem for the double rough integrals of Y via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science applications, are jointly controlled paths.
