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A Fubini type theorem for rough integration

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Title: A Fubini type theorem for rough integration
Authors: Cass, T
Pei, J
Item Type: Journal Article
Abstract: Jointly controlled paths as used in Gerasimovics and Hairer (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 Qp-rough paths X and QX (on finite dimensional spaces) for arbitrary p and Qp, and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and QX , they are integrable with respect to X and QX, 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.
Issue Date: 27-Jan-2023
Date of Acceptance: 16-Dec-2022
URI: http://hdl.handle.net/10044/1/102255
DOI: 10.4171/RMI/1409
ISSN: 0213-2230
Publisher: EMS Press
Start Page: 761
End Page: 802
Journal / Book Title: Revista Matematica Iberoamericana
Volume: 39
Issue: 2
Copyright Statement: © 2023 Real Sociedad Matemática Española Published by EMS Press and licensed under a CC BY 4.0 license (https://creativecommons.org/licenses/by/4.0/)
Notes: 40 pages
Publication Status: Published
Online Publication Date: 2023-01-27
Appears in Collections:Financial Mathematics
Faculty of Natural Sciences
Mathematics