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This paper proposes an adaptive timestep construction for an Euler-Maruyamaapproximation of the ergodic SDEs with a drift which is not globally Lipschitzover an infinite time interval. If the timestep is bounded appropriately, weshow not only the stability of the numerical solution and the standard strongconvergence order, but also that the bound for moments and strong error of thenumerical solution are uniform in T, which allow us to introduce the adaptivemultilevel Monte Carlo. Numerical experiments support our analysis.

Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval