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The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
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Boedihardjo, Horatio, Diehl, Joscha, Mezzarobba, Marc and Ni, Hao (2021) The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence. Bulletin of the London Mathematical Society . doi:10.1112/blms.12420 (In Press)
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Official URL: https://doi.org/10.1112/blms.12420
Abstract
The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science > Statistics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Laplace transformation, Probabilities, Brownian motion processes, Convergence | |||||||||||||||
| Journal or Publication Title: | Bulletin of the London Mathematical Society | |||||||||||||||
| Publisher: | Wiley | |||||||||||||||
| ISSN: | 0024-6093 | |||||||||||||||
| Official Date: | 2021 | |||||||||||||||
| Dates: |
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| Date of first compliant deposit: | 25 September 2020 | |||||||||||||||
| DOI: | 10.1112/blms.12420 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | In Press | |||||||||||||||
| Access rights to Published version: | Open Access | |||||||||||||||
| Copyright Holders: | © 2020 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society. | |||||||||||||||
| RIOXX Funder/Project Grant: |
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