Monte Carlo and Quasi-Monte Carlo Sampling (eBook)

Christiane Lemieux (Autor)

eBook Download: PDF

2009 | 2009
XIV, 373 Seiten
Springer New York (Verlag)
978-0-387-78165-5 (ISBN)

Infos

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.

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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 "e;Young Researcher Award in Information-Based Complexity"e; in 2004.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 3.4.2009
• Reihe/Serie: Springer Series in Statistics
• Reihe/Serie: Springer Series in Statistics
• Zusatzinfo: XIV, 373 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: ANOVA • Integration • Monte Carlo • Monte Carlo Method • quasi-Monte Carlo • Simulation • Statistica • Variance
• ISBN-10: 0-387-78165-X / 038778165X
• ISBN-13: 978-0-387-78165-5 / 9780387781655