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The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Articles and Preprints
We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based …
Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Articles and Preprints
We study the behavior for large |x| of Kunita-type stochastic flows φ(t, ω, x) on Rd, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large |x|, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.