Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise
- Creators
- Mattingly, J. C.
- Stuart, A. M.
- Higham, D. J.
Abstract
The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn–Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler–Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.
Additional Information
© 2002 Elsevier Science B.V. Received 8 August 2000, Revised 21 March 2002, Accepted 10 April 2002, Available online 28 May 2002. Supported by the National Science Foundation under grant DMS-9971087. Supported by the Engineering and Physical Sciences Research Council of the UK under grant GR/N00340. Supported by the Engineering and Physical Sciences Research Council of the UK under grant GR/M42206.Additional details
- Eprint ID
- 78060
- Resolver ID
- CaltechAUTHORS:20170609-125050787
- DMS-9971087
- NSF
- GR/N00340
- Engineering and Physical Sciences Research Council (EPSRC)
- GR/M42206
- Engineering and Physical Sciences Research Council (EPSRC)
- Created
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2017-06-09Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field
- Other Numbering System Name
- Andrew Stuart
- Other Numbering System Identifier
- J50