Pathwise Accuracy and 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 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.