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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, 53 (1). pp. 285-299. doi:10.1112/blms.12420 ISSN 0024-6093 . [ 🗎 Public]. [ (🔓): Yes ].

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Official URL: https://doi.org/10.1112/blms.12420

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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
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > 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: February 2021
Dates:
DateEvent
February 2021Published
28 October 2020Available
21 September 2020Accepted
Volume: 53
Number: 1
Page Range: pp. 285-299
DOI: 10.1112/blms.12420
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Copyright Holders: © 2020 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.
Date of first compliant deposit: 25 September 2020
Date of first compliant Open Access: 26 November 2020
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/R008205/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
ANR-14-CE25-0018-01[ANR] Agence Nationale de la Recherchehttp://dx.doi.org/10.13039/501100001665
EP/S026347/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
EP/N510129/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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