Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient

Thomas Müller-Gronbach and Larisa Yaroslavtseva

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Abstract

Recently a lot of effort has been invested to analyze the $L_{p}$-error of the Euler–Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_{p}$-error rate of at least $1/(2p)$ – has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler–Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2$ for all $p\in [1,\infty )$ as in the case of SDEs with Lipschitz coefficients. The proof of this result is based on a detailed analysis of appropriate occupation times for the Euler–Maruyama scheme.

Résumé

De nombreux efforts ont été consacrés récemment à l’analyse de l’erreur $L_{p}$ de schéma d’Euler–Maruyama pour des équations différentielles stochastiques (EDS) avec un coefficient de dérive pouvant avoir des discontinuités en espace. Jusqu’à présent, pour des EDS scalaires avec un coefficient de dérive Lipschitz par morceaux et un coefficient de diffusion Lipschitz qui est non nul aux points de discontinuité du coefficient de dérive, seule une borne d’erreur $L_{p}$ avec un taux d’au moins $1/(2p)$ – a été obtenue. Dans cet article, nous montrons que sous les hypothèses précédentes, le schéma d’Euler–Maruyama réalise un taux d’erreur $L_{p}$ d’au moins $1/2$ pour tout $p\in [1,\infty )$, comme dans le cas d’EDS avec coefficients Lipschitz. La preuve de ce résultat se fonde sur une analyse détaillée de temps d’occupation bien choisis pour le schéma d’Euler–Maruyama.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1162-1178.

Dates
Received: 25 September 2018
Revised: 7 March 2019
Accepted: 29 April 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345633

Digital Object Identifier
doi:10.1214/19-AIHP997

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

Keywords
Stochastic differential equations Discontinuous drift coefficient Strong approximation Euler–Maruyama scheme

Citation

Müller-Gronbach, Thomas; Yaroslavtseva, Larisa. On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1162--1178. doi:10.1214/19-AIHP997. https://projecteuclid.org/euclid.aihp/1584345633


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