The Stable Manifold Theorem for Stochastic Differential Equations

Authors

Salah-Eldin A. Mohammed, Southern Illinois University CarbondaleFollow
Michael K. R. Scheutzow, Technical University of Berlin

Comments

Published in Annals of Probability, 27(2), 615-652. Preprint posted in MSRI Preprint 1998-015

Abstract

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 on Ruelle–Oseledec multiplicative ergodic theory.

Recommended Citation

Mohammed, Salah-Eldin A. and Scheutzow, Michael K. "The Stable Manifold Theorem for Stochastic Differential Equations." (Jan 1999).