Electronic Journal of Probability

The non-linear sewing lemma I: weak formulation

Antoine Brault and Antoine Lejay

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We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even if solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 59, 24 pp.

Received: 27 February 2018
Accepted: 2 May 2019
First available in Project Euclid: 21 June 2019

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 54C65: Selections [See also 28B20]

rough paths rough differential equations non uniqueness of solutions flow approximations measurable flows Lipschitz flows sewing lemma

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Brault, Antoine; Lejay, Antoine. The non-linear sewing lemma I: weak formulation. Electron. J. Probab. 24 (2019), paper no. 59, 24 pp. doi:10.1214/19-EJP313. https://projecteuclid.org/euclid.ejp/1561082668

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