Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise
Mattingly, J. 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. 185232. ISSN 03044149

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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 MeynTweedie. 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 dampeddriven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study timediscrete 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 EulerMaruyama; 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 

ID code:  171 
Keywords:  geometric ergodicity, stochastic equations, Langevin equation, gradient systems, additive noise, timediscretization, computer science, applied mathematics, Electronic computers. Computer science, Mathematics, Modelling and Simulation, Applied Mathematics, Statistics and Probability 
Subjects:  Science > Mathematics > Electronic computers. Computer science Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Ms Sarah Scott 
Date Deposited:  03 Mar 2006 
Last modified:  27 May 2016 13:53 
URI:  http://strathprints.strath.ac.uk/id/eprint/171 