Ergodicity For SDEs and Approximations: Locally Lipschitz Vector Fields
نویسندگان
چکیده
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. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set, and some smoothness of the probability densities; the last two points imply a minorization condition. Together the minorization condition and Lyapunov structure give geometric ergodicity. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are strong, yielding exponential convergence of expectations for classes of measurable functions restricted only by the condition that they grow no faster than the Lyapunov function. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. 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; 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.
منابع مشابه
Weak differentiability of solutions to SDEs with semi-monotone drifts
In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To prove this formula, we construct a sequence of SDEs with globally Lipschitz drifts and show that the $p$-moments of their Malliavin derivatives are uniformly bounded.
متن کاملOn Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations
We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity ass...
متن کاملPathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finitetime intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized...
متن کاملPathwise Accuracy & Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDE) are proposed which (i) are ergodic with respect to the (known) equilibrium distribution of the SDE and (ii) approximate pathwise the solutions of the SDE on finite time intervals. Both these properties are demonstrated in the paper and precise strong error estimates are obtained. It is also shown that the Metropolized ...
متن کامل