Without requiring any background knowledge of stochastic differential equations (SDEs), I will aim to give an overview of some recent results concerning their numerical simulation. The results were proved in collaboration with Xuerong Mao (Strathclyde) and Andrew Stuart (Warwick), and their purpose is to get around the somewhat restrictive global Lipschitz assumption that appears in the traditional analysis of SDE methods. The key idea, borrowed from the deterministic literature, is to weaken the global Lipschitz condition on the drift coefficient to a one-sided Lipschitz (OSL) condition.

First, I will consider finite time, strong convergence. I will show that an implicit variant of the Euler-Maruyama method converges under a OSL condition on the drift. Further, the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial. I will then look at a long-time asymptotic property: the ability of numerical methods to reproduce exponential mean square stability of SDEs. A simple counterexample demonstrates that, for non-globally Lipschitz SDEs, this form of stability is not inherited, in general, by numerical methods. However, I will show that under a suitable OSL

condition, positive results may be obtained for two implicit methods.

These results emphasize that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.