Electronic Communications in Probability

Uniform Factorial Decay Estimates for Controlled Differential Equations

Horatio Boedihardjo, Terry Lyons, and Danyu Yang

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We establish an uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations in the p-variation metric. As part of the proof, we also obtain a factorial decay estimate for controlled paths which is interesting in its own right.

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Electron. Commun. Probab., Volume 20 (2015), paper no. 94, 11 pp.

Accepted: 19 December 2015
First available in Project Euclid: 7 June 2016

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Primary: Probability
Secondary: Rough path theory Controlled differential equation

Controlled differential equation Rough paths Taylor expansion Factorial Decay

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Boedihardjo, Horatio; Lyons, Terry; Yang, Danyu. Uniform Factorial Decay Estimates for Controlled Differential Equations. Electron. Commun. Probab. 20 (2015), paper no. 94, 11 pp. doi:10.1214/ECP.v20-4124. https://projecteuclid.org/euclid.ecp/1465321021

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  • Hara, Keisuke; Hino, Masanori. Fractional order Taylor's series and the neo-classical inequality. Bull. Lond. Math. Soc. 42 (2010, no. 3, 467–477.
  • H. Boedihardjo, Decay rate of iterated integrals of branched rough paths, arXiv:1501.05641, 2015.
  • Davie, A. M. Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. A 2007, no. 2, Art. ID abm009, 40 pp.
  • Friz, Peter K; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths: Theory and applications, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, Cambridge, 2010, xiv+656 pp. ISBN: 978-0-521-87607-0
  • Friz, Peter; Victoir, Nicolas, Euler estimates for rough differential equations, J. Differential Equations 244,2008, no. 2, 388–412.
  • Gubinelli, M, Controlling rough paths, J. Funct. Anal. 216,2004, no. 1, 86–140,
  • Lyons, Terry, Differential equations driven by rough signals: I An extension of an inequality of L. C. Young, Math. Res. Lett. 1 1994, no. 4, 451–464.
  • Lyons, Terry J, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 1998, no. 2, 215–310.
  • P. Yam, textitAnalytical and Topological Aspects of Signatures, D.Phil Thesis, available at http://ora.ox.ac.uk/objects/uuid:87892930-f329-4431-bcdc-bf32b0b1a7c6/datastreams/ATTACHMENT1
  • Young, L. C., An inequality of the Holder type, connected with Stieltjes integration, Acta Math. 67 1936, no. 1, 251–282.