  This paper proposes an adaptive timestep construction for an Euler-Maruyama
approximation of SDEs with a drift which is not globally Lipschitz. It is
proved that if the timestep is bounded appropriately, then over a finite time
interval the numerical approximation is stable, and the expected number of
timesteps is finite. Furthermore, the order of strong convergence is the same
as usual, i.e. order one-half for SDEs with a non-uniform globally Lipschitz
volatility, and order one for Langevin SDEs with unit volatility and a drift
with sufficient smoothness. The analysis is supported by numerical experiments
for a variety of SDEs.
