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

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