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

J. Mattingly, A.M. Stuart, D.J. Higham

Research output: Contribution to journalArticle

221 Citations (Scopus)

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.
LanguageEnglish
Pages185-232
Number of pages47
JournalStochastic Processes and their Applications
Volume101
Issue number2
DOIs
Publication statusPublished - 2002

Fingerprint

Ergodicity
Markov processes
Lipschitz
Vector Field
Lyapunov
Approximation
Markov chain
Langevin Equation
Hamiltonians
Time and motion study
Discretization
Geometric Ergodicity
Backward Euler Method
Lorenz Equations
Gradient System
Explicit Methods
Time Discretization
Damped
Euler
State Space

Keywords

  • geometric ergodicity
  • stochastic equations
  • Langevin equation
  • gradient systems
  • additive noise
  • time-discretization
  • computer science
  • applied mathematics

Cite this

Mattingly, J. ; Stuart, A.M. ; Higham, D.J. / Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. In: Stochastic Processes and their Applications. 2002 ; Vol. 101, No. 2. pp. 185-232.
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Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. / Mattingly, J.; Stuart, A.M.; Higham, D.J.

In: Stochastic Processes and their Applications, Vol. 101, No. 2, 2002, p. 185-232.

Research output: Contribution to journalArticle

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