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The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
- 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.
Author: | Horatio Boedihardjo, Joscha Diehl, Marc Mezzarobba, Hao Ni |
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URN: | urn:nbn:de:gbv:9-opus-43248 |
DOI: | https://doi.org/10.1112/blms.12420 |
Parent Title (English): | Bulletin of the London Mathematical Society |
Document Type: | Article |
Language: | English |
Date of first Publication: | 2021/02/01 |
Release Date: | 2021/02/16 |
Tag: | 33C10; 60B15; 60H05 (primary) |
GND Keyword: | - |
Volume: | 53 |
Issue: | 1 |
First Page: | 285 |
Last Page: | 299 |
Faculties: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik und Informatik |
Licence (German): | Creative Commons - Namensnennung |