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February 2019 Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients
Mario Hefter, André Herzwurm, Thomas Müller-Gronbach
Ann. Appl. Probab. 29(1): 178-216 (February 2019). DOI: 10.1214/18-AAP1411

Abstract

We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox–Ingersoll–Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases, the resulting lower error bounds even turn out to be sharp.

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Mario Hefter. André Herzwurm. Thomas Müller-Gronbach. "Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients." Ann. Appl. Probab. 29 (1) 178 - 216, February 2019. https://doi.org/10.1214/18-AAP1411

Information

Received: 1 October 2017; Revised: 1 May 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039124
MathSciNet: MR3910003
Digital Object Identifier: 10.1214/18-AAP1411

Subjects:
Primary: 60H10, 65C30

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2019
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