Adaptive Euler–Maruyama Method for SDEs with Non-globally Lipschitz Drift

Abstract

This paper, based on two main papers Fang and Giles (Adaptive Euler–Maruyama method for SDEs with non-globally Lipschitz drift: Part I, finite time interval, 2016, [2]), Fang and Giles (Adaptive Euler–Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval, 2017, [3]) which contains the full details of the literature review, numerical analysis and numerical experiments, aims to give an overview of the adaptive Euler–Maruyama method for SDEs with non-globally Lipschitz drift in a concise structure without any proof. It shows 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 \(\frac{1}{2}\) for SDEs with a non-uniform 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 measure. The analysis is supported by numerical experiments.