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Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

Mattingly, J. C. and Stuart, A. M. and Higham, D. J. (2002) Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Processes and their Applications, 101 (2). pp. 185-232. ISSN 0304-4149. doi:10.1016/S0304-4149(02)00150-3. https://resolver.caltech.edu/CaltechAUTHORS:20170609-125050787

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/S0304-4149(02)00150-3DOIArticle
http://www.sciencedirect.com/science/article/pii/S0304414902001503?via%3DihubPublisherArticle
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.
Funders:
Funding AgencyGrant Number
NSFDMS-9971087
Engineering and Physical Sciences Research Council (EPSRC)GR/N00340
Engineering and Physical Sciences Research Council (EPSRC)GR/M42206
Subject Keywords:Geometric ergodicity; Stochastic differential equations; Langevin equation; Monotone; Dissipative and gradient systems; Additive noise; Hypoelliptic and degenerate diffusions; Time-discretization
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ50
Issue or Number:2
DOI:10.1016/S0304-4149(02)00150-3
Record Number:CaltechAUTHORS:20170609-125050787
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170609-125050787
Official Citation:J.C. Mattingly, A.M. Stuart, D.J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Processes and their Applications, Volume 101, Issue 2, 2002, Pages 185-232, ISSN 0304-4149, http://dx.doi.org/10.1016/S0304-4149(02)00150-3. (http://www.sciencedirect.com/science/article/pii/S0304414902001503)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78060
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:09 Jun 2017 21:14
Last Modified:15 Nov 2021 17:36

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