The Tail Asymptotics of the Brownian Signature
Abstract
The signature of a path \gamma is a sequence whose n-th term is the order-n iterated integrals of \gamma. It arises from solving multidimensional linear differential equations driven by \gamma. We are interested in relating the path properties of \gamma with its signature. If \gamma is C^{1}, then an elegant formula of Hambly and Lyons relates the length of \gamma to the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion, with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion.
Keywords
Cite
@article{arxiv.1609.08111,
title = {The Tail Asymptotics of the Brownian Signature},
author = {Horatio Boedihardjo and Xi Geng},
journal= {arXiv preprint arXiv:1609.08111},
year = {2018}
}
Comments
Expanded on the background materials on hyperbolic development, and the upper bound now holds outside a null set independent of time