Monte Carlo Methods. Handbook of. University ofqueensland. Thomas Taimre. Zdravko I. Botev. Dirk P. Kroese. Universite de Montreal

Transcription

1 Handbook of Monte Carlo Methods Dirk P. Kroese University ofqueensland Thomas Taimre University ofqueensland Zdravko I. Botev Universite de Montreal A JOHN WILEY & SONS, INC., PUBLICATION

2 Preface Acknowledgments xvii xix 1 Uniform Random Number Generation Random Numbers Properties of a Good Random Number Generator Choosing a Good Random Number Generator Generators Based on Linear Recurrences Linear Congruential Generators Multiple-Recursive Generators Matrix Congruential Generators Modulo 2 Linear Generators Combined Generators Other Generators Tests for Random Number Generators Spectral Test Empirical Tests 14 References 21 vii

3 viii 2 Quasirandom Number Generation Multidimensional Integration Van der Corput and Digital Sequences Halton Sequences Faure Sequences 2.5 Sobol' Sequences Lattice Methods Randomization and Scrambling 38 References Random Variable Generation Generic Algorithms Based on Common Transformations Inverse-Transform Method Other Transformation Methods Table Lookup Method Alias Method Acceptance -Rejection Method Ratio of Uniforms Method Generation Methods for Multivariate Random Variables Copulas Generation Methods for Various Random Objects Generating Order Statistics Generating Uniform Random Vectors in a Simplex Generating Random Vectors Uniformly Distributed in a Unit Hyperball and Hypersphere Generating Random Vectors Uniformly Distributed in a Hyperellipsoid Uniform Sampling on a Curve Uniform Sampling on a Surface Generating Random Permutations Exact Sampling From a Conditional Bernoulli Distribution 80 References 83 4 Probability Distributions Discrete Distributions Bernoulli Distribution Binomial Distribution Geometric Distribution Hypergeometric Distribution Negative Binomial Distribution 94

4 ix Phase-Type Distribution (Discrete Case) Poisson Distribution Uniform Distribution (Discrete Case) Continuous Distributions Beta Distribution Cauchy Distribution Exponential Distribution F Distribution Frechet Distribution Gamma Distribution Gumbel Distribution Laplace Distribution Logistic Distribution Log-Normal Distribution Normal Distribution Pareto Distribution Phase-Type Distribution (Continuous Case) Stable Distribution Student's t Distribution Uniform Distribution (Continuous Case) Wald Distribution Weibull Distribution Multivariate Distributions Dirichlet Distribution Multinomial Distribution Multivariate Normal Distribution Multivariate Student's t Distribution Wishart Distribution 148 References Random Process Generation Gaussian Processes Markovian Gaussian Processes Stationary Gaussian Processes and the FFT Markov Chains Markov Jump Processes Poisson Processes Compound Poisson Process Wiener Process and Brownian Motion Stochastic Differential Equations and Diffusion Processes Euler's Method Milstein's Method 187

5 X Implicit Euler Exact Methods Error and Accuracy Brownian Bridge Geometric Brownian Motion Ornstein-Uhlenbeek Process Reflected Brownian Motion Fractional Brownian Motion Random Fields Levy Processes Increasing Levy Processes Generating Levy Processes Time Series 219 References Markov Chain Monte Carlo Metropolis-Hastings Algorithm Independence Sampler Random Walk Sampler Gibbs Sampler Specialized Samplers Hit-and-Rim Sampler Shake-and-Bake Sampler Metropolis-Gibbs Hybrids Multiple-Try Metropolis-Hastings Auxiliary Variable Methods Reversible Jump Sampler Implementation Issues Perfect Sampling 274 References Discrete Event Simulation Simulation Models Discrete Event Systems Event-Oriented Approach More Examples of Discrete Event Simulation Inventory System Tandem Queue Repairman Problem 296 References 300

6 Xi 8 Statistical Analysis of Simulation Data Simulation Data Data Visualization Data Summarization Estimation of Performance Measures for Independent Data Delta Method Estimation of Steady-State Performance Measures Covariance Method Batch Means Method Regenerative Method Empirical Cdf Kernel Density Estimation Least Squares Cross Validation Plug-in Bandwidth Selection Resampling and the Bootstrap Method Goodness of Fit Graphical Procedures Kolmogorov-Smirnov Test Anderson-Darling Test x2 Tests 340 References Variance Reduction Variance Reduction Example Antithetic Random Variables Control Variables Conditional Monte Carlo Stratified Sampling Latin Hypercube Sampling Importance Sampling Minimum-Variance Density Variance Minimization Method Cross-Entropy Method Weighted Importance Sampling Sequential Importance Sampling Response Surface Estimation via Importance Sampling Quasi Monte Carlo 376 References 379

7 Xii 10 Rare-Event Simulation Efficiency of Estimators Importance Sampling Methods for Light Tails Estimation of Stopping Time Probabilities Estimation of Overflow Probabilities Estimation For Compound Poisson Sums Conditioning Methods for Heavy Tails Estimation for Compound Sums Sum of Nonidentically Distributed Random Variables State-Dependent Importance Sampling Cross-Entropy Method for Rare-Event Simulation Splitting Method 409 References Estimation of Derivatives Gradient Estimation Finite Difference Method Infinitesimal Perturbation Analysis Score Function Method Score Function Method With Importance Sampling Weak Derivatives Sensitivity Analysis for Regenerative Processes 435 References Randomized Optimization Stochastic Approximation Stochastic Counterpart Method Simulated Annealing Evolutionary Algorithms Genetic Algorithms Differential Evolution Estimation of Distribution Algorithms Cross-Entropy Method for Optimization Other Randomized Optimization Techniques 460 References Cross-Entropy Method Cross-Entropy Method Cross-Entropy Method for Estimation Cross-Entropy Method for Optimization Combinatorial Optimization 469

8 Xlii Continuous Optimization Constrained Optimization Noisy Optimization 476 References Particle Methods Sequential Monte Carlo Particle Splitting Splitting for Static Rare-Event Probability Estimation Adaptive Splitting Algorithm Estimation of Multidimensional Integrals Combinatorial Optimization via Splitting Knapsack Problem Traveling Salesman Problem Quadratic Assignment Problem Markov Chain Monte Carlo With Splitting 509 References Applications to Finance Standard Model Pricing via Monte Carlo Simulation Sensitivities Pathwise Derivative Estimation Score Function Method 542 References Applications to Network Reliability Network Reliability Evolution Model for a Static Network Conditional Monte Carlo Leap-Evolve Algorithm Importance Sampling for Network Reliability Importance Sampling Using Bounds Importance Sampling With Conditional Monte Carlo Splitting Method Acceleration Using Bounds 573 References Applications to Differential Equations Connections Between Stochastic and Partial Differential Equations 577

9 Xiv Boundary Value Problems Terminal Value Problems Terminal Boundary Problems Transport Processes and Equations Application to Transport Equations Boltzmann Equation Connections to ODEs Through Scaling 597 References 602 Appendix A: Probability and Stochastic Processes 605 A.l Random Experiments and Probability Spaces 605 A.1.1 Properties of a Probability Measure 607 A.2 Random Variables and Probability Distributions 607 A.2.1 Probability Density 610 A.2.2 Joint Distributions 611 A.3 Expectation and Variance 612 A.3.1 Properties of the Expectation 614 A.3.2 Variance 615 A.4 Conditioning and Independence 616 A.4.1 Conditional Probability 616 A.4.2 Independence 616 A.4.3 Covariance 617 A.4.4 Conditional Density and Expectation 618 A.5 V Space 619 A.6 Functions of Random Variables 620 A.6.1 Linear Transformations 620 A.6.2 General Transformations 620 A.7 Generating Function and Integral Transforms 621 A. 7.1 Probability Generating Function 621 A.7.2 Moment Generating Function and Laplace Transform 621 A.7,3 Characteristic Function 622 A.8 Limit Theorems 623 A.8.1 Modes of Convergence 623 A.8.2 Converse Results on Modes of Convergence 624 A.8.3 Law of Large Numbers and Central Limit Theorem 625 A.9 Stochastic Processes 626 A.9.1 Gaussian Property 627 A.9.2 Markov Property 628 A.9.3 Martingale Property 629 A.9.4 Regenerative Property 630 A.9.5 Stationarity and Reversibility 631 A. 10 Markov Chains 632

10 XV A Classification of States 633 A Limiting Behavior 633 A.10.3 Reversibility 635 A. 11 Markov Jump Processes 635 A Limiting Behavior 638 A. 12 Ito Integral and Ito Processes 639 A. 13 Diffusion Processes 643 A.13.1 Kolrnogorov Equations 646 A.13.2 Stationary Distribution 648 A Feynman-Kac Formula 648 A Exit Times 649 References 650 Appendix B: Elements of Mathematical Statistics 653 B. l Statistical Inference 653 B. l.l Classical Models 654 B.l.2 Sufficient Statistics 655 B.l. 3 Estimation 656 B.1.4 Hypothesis Testing 660 B.2 Likelihood 664 B.2.1 Likelihood Methods for Estimation 667 B.2.2 Numerical Methods for Likelihood Maximization 669 B.2.3 Likelihood Methods for Hypothesis Testing 671 B. 3 Bayesian Statistics 672 B. 3.1 Conjugacy 675 References 676 Appendix C: Optimization 677 C. l Optimization Theory 677 C. l.l Lagrangian Method 683 C.1.2 Duality 684 C.2 Techniques for Optimization 685 C.2.1 Transformation of Constrained Problems 685 C.2.2 Numerical Methods for Optimization and Root Finding 687 C.3 Selected Optimization Problems 694 C.3.1 Satisfiability Problem 694 C.3.2 Knapsack Problem 694 C.3.3 Max-Cut Problem 695 C.3.4 Traveling Salesman Problem 695 C.3.5 Quadratic Assignment Problem 695 C.3.6 Clustering Problem 696

11 xvi C. 4 Continuous Problems 696 C.4.1 Unconstrained Problems 696 C. 4.2 Constrained Problems 697 References 699 Appendix D: Miscellany 701 D. l Exponential Families 701 D.2 Properties of Distributions 703 D. 2.1 Tail Properties 703 D.2.2 Stability Properties 705 D.3 Cholesky Factorization 706 D.4 Discrete Fourier Transform, FFT, and Circulant Matrices 706 D.5 Discrete Cosine Transform 708 D.6 Differentiation 709 D.7 Expectation-Maximization (EM) Algorithm 711 D.8 Poisson Summation Formula 714 D.9 Special Functions 715 D.9.1 Beta Function B{a,(3) 715 D.9.2 Incomplete Beta Function Ix(a, (i) 715 D.9.3 Error Function erf(x) 715 D.9.4 Digamma function 4>{'x) 716 D.9.5 Gamma Function T{a) 716 D.9.6 Incomplete Gamma Function P(a, x) 716 D.9.7 Hypergeometric Function 2Fi(a, 6; c; 2) 716 D.9.8 Confluent Hypergeometric Function i-fi(a; 7; j:) 717 D.9.9 Modified Bessel Function of the Second Kind K {x) 717 References 717 Acronyms and Abbreviations 719 List of Symbols 721 List of Distributions 724 Index 727