Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs

TL;DR

This paper introduces Metropolized integrators for ergodic SDEs that ensure ergodicity and pathwise accuracy, providing strong error estimates and stability even with non-globally Lipschitz drifts.

Contribution

It develops and analyzes Metropolized integrators that preserve ergodicity and pathwise accuracy for SDEs, including challenging cases with non-globally Lipschitz drifts.

Findings

Metropolized integrators are ergodic with respect to the SDE's equilibrium distribution.

They approximate SDE solutions pathwise on finite intervals.

The integrators remain stable and ergodic even with non-globally Lipschitz drifts.

Abstract

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 integrator retains these properties even in situations where the drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for SDEs typically become unstable and fail to be ergodic.