Lemieux Monte Carlo and Quasi-Monte Carlo Sampling

1. Auflage 2009
ISBN: 978-0-387-78165-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 382 Seiten

Reihe: Springer Series in Statistics

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 areas within this field to which practitioners and researchers from various disciplines currently contribute. This book presents essential tools for using quasi-Monte Carlo sampling in practice. The first part of the book focuses on issues related to Monte Carlo methods-uniform and non-uniform random number generation, variance reduction techniques-but the material is presented to prepare the readers for the next step, which is to replace the random sampling inherent to Monte Carlo by quasi-random sampling. The second part of the book deals with this next step. Several aspects of quasi-Monte Carlo methods are covered, including constructions, randomizations, the use of ANOVA decompositions, and the concept of effective dimension. The third part of the book is devoted to applications in finance and more advanced statistical tools like Markov chain Monte Carlo and sequential Monte Carlo, with a discussion of their quasi-Monte Carlo counterpart. The prerequisites for reading this book are a basic knowledge of statistics and enough mathematical maturity to follow through the various techniques used throughout the book. This text is aimed at graduate students in statistics, management science, operations research, engineering, and applied mathematics. It should also be useful to practitioners who want to learn more about Monte Carlo and quasi-Monte Carlo methods and researchers interested in an up-to-date guide to these methods.

Christiane Lemieux is an Associate Professor and the Associate Chair for Actuarial Science in the Department of Statistics and Actuarial Science at the University of Waterloo in Canada. She is an Associate of the Society of Actuaries and was the winner of a 'Young Researcher Award in Information-Based Complexity' in 2004.

Autoren/Hrsg.

Lemieux, Christiane

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Acronyms and Symbols;12
4;Chapter 1 The Monte Carlo Method;14
4.1;1.1 Monte Carlo method for integration;16
4.2;1.2 Connection with stochastic simulation;25
4.3;1.3 Alternative formulation of the integration problem via f: an example;33
4.4;1.4 A primer on uniform random number generation;35
4.5;1.5 Using Monte Carlo to approximate a distribution;38
4.6;1.6 Two more examples;40
4.7;Problems;47
5;Chapter 2 Sampling from Known Distributions;53
5.1;2.1 Common distributions arising in stochastic models;54
5.2;2.2 Inversion;56
5.3;2.3 Acceptance-rejection;58
5.4;2.4 Composition;60
5.5;2.5 Convolution and other useful identities;62
5.6;2.6 Multivariate case;63
5.7;Problems;67
6;Chapter 3 Pseudorandom Number Generators;69
6.1;3.1 Basic concepts and definitions;70
6.2;3.2 Generators based on linear recurrences;72
6.3;3.3 Add-with-carry and subtract-with-borrow generators;78
6.4;3.4 Nonlinear generators;79
6.5;3.5 Theoretical and statistical testing;80
6.6;Problems;97
7;Chapter 4 Variance Reduction Techniques;99
7.1;4.1 Introduction;99
7.2;4.2 Efficiency;101
7.3;4.3 Antithetic variates;101
7.4;4.4 Control variates;113
7.5;4.5 Importance sampling;123
7.6;4.6 Conditional Monte Carlo;131
7.7;4.7 Stratification;137
7.8;4.8 Common random numbers;144
7.9;4.9 Combinations of techniques;147
7.10;Problems;148
8;Chapter 5 Quasi–Monte Carlo Constructions;151
8.1;5.1 Introduction;151
8.2;5.2 Main constructions: basic principles;155
8.3;5.3 Lattices;158
8.4;5.4 Digital nets and sequences;165
8.5;5.5 Recurrence-based point sets;186
8.6;5.6 Quality measures;191
8.7;Problems;209
9;Chapter 6 Using Quasi–Monte Carlo in Practice;212
9.1;6.1 Introduction;212
9.2;6.2 Randomized quasi–Monte Carlo;213
9.3;6.3 ANOVA decomposition and effective dimension;225
9.4;6.4 Using quasi–Monte Carlo sampling for simulation;240
9.5;6.5 Suggestions for practitioners;248
9.6;Problems;250
9.7;Appendix: Tractability, weighted spaces, and component-by-component constructions;252
10;Chapter 7 Financial Applications;258
10.1;7.1 European option pricing under the lognormal model;258
10.2;7.2 More complex models;267
10.3;7.3 Randomized quasi–Monte Carlo methods in finance;271
10.4;7.4 Commonly used variance reduction techniques;284
10.5;7.5 American option pricing;294
10.6;7.6 Estimating sensitivities and percentiles;299
10.7;Problems;309
11;Chapter 8 Beyond Numerical Integration;312
11.1;8.1 Markov Chain Monte Carlo (MCMC);314
11.2;8.2 Sequential Monte Carlo;323
11.3;8.3 Computer experiments;331
11.4;Problems;343
12;Appendix A Review of Algebra;345
13;Appendix B Error and Variance Analysis for Halton Sequences;350
14;References;355
15;Index;377