Electronic Journal of Probability

Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

Benjamin Jourdain and Ahmed Kebaier

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Abstract

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 12, 34 pp.

Dates
Received: 4 September 2017
Accepted: 23 January 2019
First available in Project Euclid: 20 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1550653271

Digital Object Identifier
doi:10.1214/19-EJP271

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30] 65C30: Stochastic differential and integral equations 65C05: Monte Carlo methods 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
non asymptotic bounds Euler scheme multilevel Monte Carlo methods Malliavin calculus

Rights
Creative Commons Attribution 4.0 International License.

Citation

Jourdain, Benjamin; Kebaier, Ahmed. Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient. Electron. J. Probab. 24 (2019), paper no. 12, 34 pp. doi:10.1214/19-EJP271. https://projecteuclid.org/euclid.ejp/1550653271


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References

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