## Annals of Applied Probability

### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 30, Number 2 (2020), 526-560.

Dates
First available in Project Euclid: 8 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1591603215

Digital Object Identifier
doi:10.1214/19-AAP1507

Mathematical Reviews number (MathSciNet)
MR4108115

#### Citation

Fang, Wei; Giles, Michael B. Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift. Ann. Appl. Probab. 30 (2020), no. 2, 526--560. doi:10.1214/19-AAP1507. https://projecteuclid.org/euclid.aoap/1591603215

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