@article{BoedihardjoDiehlMezzarobbaetal.2021,
  author    = {Horatio Boedihardjo and Joscha Diehl and Marc Mezzarobba and Hao Ni},
  title     = {The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence},
  series   = {Bulletin of the London Mathematical Society},
  volume    = {53},
  number    = {1},
  doi       = {10.1112/blms.12420},
  url       = {https://nbn-resolving.org/urn:nbn:de:gbv:9-opus-43248},
  pages     = {285 -- 299},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}