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Variance reduction via basis expansion in Monte Carlo integration

Yazhen Wang

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Abstract

Monte Carlo methods are widely used in numerical integration, and variance reduction plays a key role in Monte Carlo integration. This paper investigates variance reduction for Monte Carlo integration in both finite dimensional Euclidean space and infinite dimensional Wiener space. The proposed variance reduction approaches are to use basis functions to construct control variates for finite dimensional integrals and utilize Itô-Wiener chaos expansion to design antithetic variates and control variates for Wiener integrals. We establish the variances of the proposed Monte Carlo integration estimators and show that the proposed methods can achieve dramatic variance reduction in comparison with the basic Monte Carlo estimators. Examples are used to illustrate the performance of the proposed estimators.

Chapter information

Source
James O. Berger, T. Tony Cai and Iain M. Johnstone, eds., Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2010), 234-248

Dates
First available in Project Euclid: 26 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1288099023

Digital Object Identifier
doi:10.1214/10-IMSCOLL616

Mathematical Reviews number (MathSciNet)
MR2798522

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 62G05: Estimation 65C30: Stochastic differential and integral equations

Keywords
antithetic variates control variates estimator Itô-Wiener chaos expansion orthonormal basis simulation Wiener process

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Wang, Yazhen. Variance reduction via basis expansion in Monte Carlo integration. Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown, 234--248, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2010. doi:10.1214/10-IMSCOLL616. https://projecteuclid.org/euclid.imsc/1288099023


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