The Annals of Probability

The Stable Manifold Theorem for Stochastic Differential Equations

Salah-Eldin A. Mohammed and Michael K. R. Scheutzow

Full-text: Open access

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 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.

Article information

Source
Ann. Probab., Volume 27, Number 2 (1999), 615-652.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677380

Digital Object Identifier
doi:10.1214/aop/1022677380

Mathematical Reviews number (MathSciNet)
MR1698943

Zentralblatt MATH identifier
0940.60084

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations
Secondary: 60H25 60H05.

Keywords
Stochastic flow spatial semimartingale local characteristics stochastic differential equation (SDE) (perfect) cocycle Lyapunov exponents hyperbolic stationary trajectory local stable/unstable manifolds asymptotic invariance

Citation

Mohammed, Salah-Eldin A.; Scheutzow, Michael K. R. The Stable Manifold Theorem for Stochastic Differential Equations. Ann. Probab. 27 (1999), no. 2, 615--652. doi:10.1214/aop/1022677380. https://projecteuclid.org/euclid.aop/1022677380


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