The Stable Manifold Theorem for Stochastic Differential Equations

Mohammed, Salah-Eldin A. ; Scheutzow, Michael K. R.

Ann. Probab., Tome 27 (1999) no. 1, p. 615-652 / Harvested from Project Euclid

Résumé

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 Itôtype 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.

Détail

Publié le : 1999-04-15
Classification:  Stochastic flow,  spatial semimartingale,  local characteristics,  stochastic differential equation (SDE),  (perfect) cocycle,  Lyapunov exponents,  hyperbolic stationary trajectory,  local stable/unstable manifolds,  asymptotic invariance,  60H10,  60H20,  60H25,  60H05.