ABSTRACT:

Numerical simulations of stochastic differential differential equations if ubiquities in modern scientific investigations. However, unlike the simulations of nonrandom differential equations, one is often interested in the statistics of long time simulations rather than the precise solution to a certain initial value problem with a given realization of forcing.

In short, the ideas of stability and consistency appropriate for a stochastic setting can be very different than in a deterministic setting. A numerical method used to simulate a stochastic differential equation amounts to a system of iterated random maps. I will describe to numerical schemes which one can prove are statistically stable if the underlying SDE is. Namely the schemes have a unique attracting invariant measure. I will also comment on when the schemes are consistent; that is there invariant measures are close to the SDEs invariant measure. In particular, I will discuss the ergodic properties of some implicit and adaptive methods.