Annals of Applied Probability

Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

Wei Fang and Michael B. Giles

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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
Received: November 2018
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

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30] 65C30: Stochastic differential and integral equations

Keywords
SDE Euler–Maruyama strong convergence adaptive time-step ergodicity invariant measure

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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