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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

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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
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:
DateEvent
15 April 2020Published
13 February 2020Available
3 February 2020Accepted
Date of first compliant deposit: 22 July 2020
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:
Project/Grant IDRIOXX Funder NameFunder ID
EP/R008205/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
DMS1814147National Science Foundationhttp://dx.doi.org/10.13039/501100008982

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