# Tail asymptotics of the Brownian signature

Boedihardjo, H. and Geng, X. (2019) Tail asymptotics of the Brownian signature. Transactions of the American Mathematical Society, 372. pp. 585-614. ISSN 1088-6850

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To link to this item DOI: 10.1090/tran/7683

## Abstract/Summary

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 C1, 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.

Item Type: Article Yes Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 78800 American Mathematical Society