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This paper proposes an adaptive timestep construction for an Euler-Maruyamaapproximation of SDEs with a drift which is not globally Lipschitz. It isproved that if the timestep is bounded appropriately, then over a finite timeinterval the numerical approximation is stable, and the expected number oftimesteps is finite. Furthermore, the order of strong convergence is the sameas usual, i.e. order one-half for SDEs with a non-uniform globally Lipschitzvolatility, and order one for Langevin SDEs with unit volatility and a driftwith sufficient smoothness. The analysis is supported by numerical experimentsfor a variety of SDEs.

Adaptive Euler-Maruyama Method for SDEs with Non-globally Lipschitz Drift: Part I, Finite Time Interval