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Tail asymptotics of the Brownian signature

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Boedihardjo, Horatio and Geng, Xi (2019) Tail asymptotics of the Brownian signature. Transactions of the American Mathematical Society, 372 (1). pp. 585-614. doi:10.1090/tran/7683

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Official URL: https://doi.org/10.1090/tran/7683

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Differential equations -- Asymptotic theory, Stochastic differential equations, Stochastic processes, Brownian motion processes
Journal or Publication Title: Transactions of the American Mathematical Society
Publisher: American Mathematical Society
ISSN: 0002-9947
Official Date: 12 April 2019
Dates:
DateEvent
12 April 2019Published
Volume: 372
Number: 1
Page Range: pp. 585-614
DOI: 10.1090/tran/7683
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: 2019 American Mathematical Society

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