Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

Abstract

This paper proposes an adaptive timestep construction for an Euler–Maruyama approximation of SDEs with nonglobally Lipschitz drift. 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, that is, order $\frac{1}{2}$ for SDEs with a nonuniform globally Lipschitz volatility, and order $1$ for Langevin SDEs with unit volatility and a drift with sufficient smoothness. For a class of ergodic SDEs, we also show that the bound for the moments and the strong error of the numerical solution are uniform in $T$, which allow us to introduce the adaptive multilevel Monte Carlo method to compute the expectations with respect to the invariant distribution. The analysis is supported by numerical experiments.