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The non-linear sewing lemma III: Stability and generic properties

The non-linear sewing lemma III: Stability and generic properties AbstractSolutions of Rough Differential Equations (RDE) may be defined as pathswhose increments are close to an approximation of the associated flow. Theyare constructed through a discrete scheme using a non-linear sewing lemma.In this article, we show that such solutions also solve a fixed pointproblem by exhibiting a suitable functional. Convergence then follows fromconsistency and stability, two notions that are adapted to our framework.In addition, we show that uniqueness and convergence of discreteapproximations is a generic property, meaning that it holds excepted for aset of vector fields and starting points which is of Baire first category.At last, we show that Brownian flows are almost surely unique solutions toRDE associated to Lipschitz flows. The later property yields almostsure convergence of Milstein schemes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

The non-linear sewing lemma III: Stability and generic properties

Forum Mathematicum , Volume 32 (5): 21 – Sep 1, 2020

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0309
Publisher site
See Article on Publisher Site

Abstract

AbstractSolutions of Rough Differential Equations (RDE) may be defined as pathswhose increments are close to an approximation of the associated flow. Theyare constructed through a discrete scheme using a non-linear sewing lemma.In this article, we show that such solutions also solve a fixed pointproblem by exhibiting a suitable functional. Convergence then follows fromconsistency and stability, two notions that are adapted to our framework.In addition, we show that uniqueness and convergence of discreteapproximations is a generic property, meaning that it holds excepted for aset of vector fields and starting points which is of Baire first category.At last, we show that Brownian flows are almost surely unique solutions toRDE associated to Lipschitz flows. The later property yields almostsure convergence of Milstein schemes.

Journal

Forum Mathematicumde Gruyter

Published: Sep 1, 2020

References