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A Fubini type theorem for rough integration
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9208287-10.4171-rmi-1409-print.pdf | Published version | 581.27 kB | Adobe PDF | View/Open |
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 |