Refine
Year of publication
- 2021 (1)
Document Type
- Article (1)
Language
- English (1)
Has Fulltext
- yes (1)
Is part of the Bibliography
- no (1)
Keywords
- 60B15 (1) (remove)
Institute
- Institut für Mathematik und Informatik (1) (remove)
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.