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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
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: |
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| 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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