Monte Carlo and Quasi-Monte Carlo Sampling
Quasi–Monte Carlo methods have become an increasingly popular alternative to Monte Carlo methods over the last two decades. Their successful implementation on practical problems, especially in finance, has motivated the development of several new research
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Christiane Lemieux
Monte Carlo and Quasi-Monte Carlo Sampling
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Christiane Lemieux University of Waterloo Department of Statistics & Actuarial Science 200 University Avenue W. Waterloo ON N2L 3G1 Canada [email protected]
ISSN: 0172-7397 ISBN: 978-0-387-78164-8 e-ISBN: 978-0-387-78165-5 DOI: 10.1007/978-0-387-78165-5 Library of Congress Control Number: 2008942366 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
A mes parents, Lise et Vincent Lemieux
Preface
The goal of this text is to provide a self-contained guide to Monte Carlo and quasi–Monte Carlo sampling methods. These two classes of methods are based on the idea of using sampling to study mathematical problems for which analytical solutions are unavailable. More precisely, the idea is to create samples that can be used to derive approximations about a quantity of interest and its probability distribution. In the former case, random sampling is used, while in the latter, low-discrepancy sampling is used. Quasi–Monte Carlo sampling methods are typically used to provide approximations for multivariate integration problems defined over the unit hypercube. They do so by creating sets or sequences of vectors (u1 , . . . , us ), with each uj taking values between 0 and 1, that sample the s-dimensional unit hypercube more regularly than random samples do, hence mimicking in a better way — with less discrepancy — the uniform distribution over that space. For this reason, most of the theory that underlies these constructions has been developed for problems that can be described as integration problems over the s-dimensional unit hypercube. On the other hand, random sampling — via the use of Monte Carlo methods — has been developed and used in a variety of situations that do not necessarily fit the formulation above, which makes use of a function defined over the unit hypercube. In particular, stochastic simulation models are usually constructed using random variables defined over the real numbers, the nonnegative integers, or other domains that are not necessarily the unit interval between 0 and 1. However, the computer implementation of such models always relies, at its lowest level, on a source of (pseudo)random nu
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