Stochastic Simulation

Machine derived contents note: 2007926471 -- Preface v -- Notation xii -- I What This Book Is About 1 -- 1 An Illustrative Example: The Single-Server Queue 1. 1 -- 2 The Monte Carlo Method 5 -- 3 Second Example: Option Pricing 6 -- 4 Issues Arising in the Monte Carlo Context 9 -- 5 Further Examples 13 -- 6 Introductory Exercises. 25 -- Part A: General Methods and Algorithms 29 -- II Generating Random Objects 30 -- 1 Uniform Random Variables 30 -- 2 Nonuniform Random Variables 36 -- 3 Multivariate Random Variables 49 -- 4 Simple Stochastic Processes 59 -- 5 Further Selected Random Objects 62 -- 6 Discrete-Event Systems and GSMPs 65 -- III Output Analysis 68 -- 1 Normal Confidence Intervals 68 -- 2 Two-Stage and Sequential Procedures 71 -- 3 Computing Smooth Functions of Expectations 73 -- 4 Computing Roots of Equations Defined by Expectations 77 -- 5 Sectioning, Jackknifing, and Bootstrapping 80 -- 6 Variance/Bias Trade-Off Issues 86 -- 7 Multivariate Output Analysis 88 -- 8 Small-Sample Theory 90 -- 9 Simulations Driven by Empirical Distributions 91 -- 10 The Simulation Budget 93 -- IV Steady-State Simulation 96 -- 1 Introduction 96 -- 2 Formulas for the Bias and Variance 102 -- 3 Variance Estimation for Stationary Processes 104 -- 4 The Regenerative Method 105 -- 5 The Method of Batch Means 109 -- 6 Further Refinements 1 110 -- 7 Duality Representations 118 -- 8 Perfect Sampling 120 -- V Variance-Reduction Methods 126 -- i Importance Sampling 127 -- 2 Control Variates 138 -- 3 Antithetic Sampling. 144 -- 4 Conditional Monte Carlo 145 -- 5 Splitting 147 -- 6 Common Random Numbers 149 -- 7 Stratification. 150 -- 8 Indirect Estimation 155 -- VI Rare-Event Simulation 158 -- 1 Efficiency Issues 158 -- 2 Examples of Efficient Algorithms: Light Tails 163 -- 3 Examples of Efficient Algorithms: Heavy Tails 173 -- 4 Tail Estimation 178 -- 5 Conditioned Limit Theorem 183 -- 6 Large-Deviations or Optimal-Path Approach 187 -- 7 Markov Chains and the h-Transform 190 -- 8 Adaptive Importance Sampling via the Cross-Entropy -- Method 195 -- 9 Multilevel Splitting 201 -- VII Derivative Estimation 206 -- 1 Finite Differences 209 -- 2 Infinitesimal Perturbation Analysis 214 -- 3 The Likelihood Ratio Method: Basic Theory 220 -- 4 The Likelihood Ratio Method: Stochastic Processes 224 -- 5 Examples and Special Methods 231 -- VIII Stochastic Optimization 242 -- 1 Introduction 242 -- 2 Stochastic Approximation Algorithms 243 -- 3 Convergence Analysis 245 -- 4 Polyak-Ruppert Averaging 250 -- 5 Examples. 253 -- Part B: Algorithms for Special Models 259 -- IX Numerical Integration 260 -- 1 Numerical Integration in One Dimension 260 -- "2 Numerical Integration in Higher Dimensions 263 -- 3 Quasi-Monte Carlo Integration 265 -- X Stochastic Differential Equations 274 -- 1 Generalities about Stochastic Process Simulation 2 274 -- 2 Brownian Motion 276 -- 3 The Euler Scheme for SDEs 280 -- 4 The Milstein and Other Higher-Order Schemes 287 -- 5 Convergence Orders for SDEs: Proofs 9292 -- 6 Approximate Error Distributions for SDEs 298 -- 7 Multidimensional SDEs 300 -- 8 Reflected Diffusions 301 -- XI Gaussian Processes 306 -- 1 Introduction 306 -- 2 Cholesky Factorization. Prediction 311 -- 3 Circulant-Embeddings 314 -- 4 Spectral Simulation. FFT 316 -- 5 Further Algorithms 320 -- 6 Fractional Brownian Motion 321 -- XII Levy Processes 325 -- 1 Introduction 325 -- 2 First Remarks on Simulation 331 -- 3 Dealing with the Small Jumps 334 -- 4 Series Representations 338 -- 3 Subordination 343 -- 6 Variance Reduction 344 -- 7 The Multidimensional Case 346 -- 8 LUvy-Driven SDEs 348 -- XIII Markov Chain Monte Carlo Methods 350 -- 1 Introduction 350 -- 2 Application Areas 352 -- 3 The Metropolis-Hastings Algorithm 361 -- 4 Special Samplers 367 -- 5 The Gibbs Sampler 375 -- XIV Selected Topics and Extended Examples 381 -- 1 Randomized Algorithms for Deterministic Optimization 381 -- 2 Resampling and Particle Filtering 385 -- 3 Counting and Measuring 391 -- 4 MCMC for the Ising Model and Square Ice 395 -- 5 Exponential Change of Measure in Markov-Modulated -- Models 403 -- 6 Further Examples of Change of Measure 407 -- 7 Black-Box Algorithms 416 -- 8 Perfect Sampling of Regenerative Processes 420 -- 9 Parallel Simulation 424 -- 10 Branching Processes 426 -- 11 Importance Sampling for Portfolio VaR 432 -- 12 Importance Sampling for Dependability Models 435 -- 13 Special Algorithms for the GI/G 1 Queue 437 -- Appendix 442 -- Al Standard Distributions 442 -- A2 Some Central Limit Theory 444 -- A3 FFT 444 -- A4 The EM Algorithm 445 -- A5 Filtering 447 -- A6 Ito's Formula 448 -- A7 Inequalities. 450 -- A8 Integral Formulas 450 -- Bibliography 452 -- Web Links 469.