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Path developments and tail asymptotics of signature for pure rough paths
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Boedihardjo, Horatio, Geng, Xi and Souris, Nikolaos P. (2020) Path developments and tail asymptotics of signature for pure rough paths. Advances in Mathematics, 364 . 107043. doi:10.1016/j.aim.2020.107043
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Official URL: http://dx.doi.org/10.1016/j.aim.2020.107043
Abstract
Solutions to linear controlled differential equations can be expressed in terms of global iterated path integrals along the driving path. This collection of iterated integrals encodes essentially all information about the underlying path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some types of asymptotics for the n-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length for paths.
In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the underlying path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science > Statistics | |||||||||
| Library of Congress Subject Headings (LCSH): | Differential equations -- Asymptotic theory, Stochastic differential equations | |||||||||
| Journal or Publication Title: | Advances in Mathematics | |||||||||
| Publisher: | Academic Press | |||||||||
| ISSN: | 0001-8708 | |||||||||
| Official Date: | 15 April 2020 | |||||||||
| Dates: |
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| Volume: | 364 | |||||||||
| Article Number: | 107043 | |||||||||
| DOI: | 10.1016/j.aim.2020.107043 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Publisher Statement: | © 2020, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/. | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| RIOXX Funder/Project Grant: |
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