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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.
Forum Mathematicum – de Gruyter
Published: Sep 1, 2020
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