TY  - JOUR
U1  - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1  - Boedihardjo, Horatio
A1  - Diehl, Joscha
A1  - Mezzarobba, Marc
A1  - Ni, Hao
T1  - The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
JF  - Bulletin of the London Mathematical Society
N2  - 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.
KW  - -
KW  - 33C10
KW  - 60B15
KW  - 60H05 (primary)
Y1  - 2021
UN  - https://nbn-resolving.org/urn:nbn:de:gbv:9-opus-43248
U6  - https://doi.org/10.1112/blms.12420
DO  - https://doi.org/10.1112/blms.12420
VL  - 53
IS  - 1
SP  - 285
EP  - 299
ER  -