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

Antoine Brault 1 Antoine Lejay 2, 3
3 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : Solutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article, we show that such solutions also solve a fixed point problem by exhibiting a suitable functional. Convergence then follows from consistency and stability, two notions that are adapted to our framework. In addition, we show that uniqueness and convergence of discrete approximations is a generic property, meaning that it holds excepted for a set of vector fields and starting points which is of Baire first category. At last, we show that Brownian flows are almost surely unique solutions to RDE associated to Lipschitz flows. The later property yields almost sure convergence of Milstein schemes.
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Antoine Brault, Antoine Lejay. The non-linear sewing lemma III: Stability and generic properties. 2019. ⟨hal-02265268⟩

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