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

TL;DR

This paper introduces an adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift, ensuring stability and convergence over finite intervals with finite expected steps, supported by numerical experiments.

Contribution

It develops an adaptive timestep scheme for SDEs with non-globally Lipschitz drift, proving stability and convergence properties that improve upon fixed timestep methods.

Findings

Stable approximation over finite intervals with bounded timesteps.

Order of strong convergence matches classical results.

Numerical experiments confirm theoretical results.

Abstract

This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of SDEs with a drift which is not globally Lipschitz. 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, i.e. order one-half for SDEs with a non-uniform globally Lipschitz volatility, and order one for Langevin SDEs with unit volatility and a drift with sufficient smoothness. The analysis is supported by numerical experiments for a variety of SDEs.