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The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

Boedihardjo, H; Diehl, J; Mezzarobba, M; Ni, H; (2020) The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence. Bulletin of the London Mathematical Society 10.1112/blms.12420. (In press). Green open access

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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.

Type: Article
Title: The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
Open access status: An open access version is available from UCL Discovery
DOI: 10.1112/blms.12420
Publisher version: https://doi.org/10.1112/blms.12420
Language: English
Additional information: This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. http://creativecommons.org/licenses/by/4.0/
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10110872
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