# Handbook of Monte Carlo Methods

## Books A comprehensive overview of Monte Carlo simulation that explores the latest topics, techniques, and real-world applications

More and more of today’s numerical problems found in engineering and finance are solved through Monte Carlo methods. The heightened popularity of these methods and their continuing development makes it important for researchers to have a comprehensive understanding of the Monte Carlo approach. Handbook of Monte Carlo Methods provides the theory, algorithms, and applications that helps provide a thorough understanding of the emerging dynamics of this rapidly-growing field.

The authors begin with a discussion of fundamentals such as how to generate random numbers on a computer. Subsequent chapters discuss key Monte Carlo topics and methods, including:

• Random variable and stochastic process generation
• Markov chain Monte Carlo, featuring key algorithms such as the Metropolis-Hastings method, the Gibbs sampler, and hit-and-run
• Discrete-event simulation
• Techniques for the statistical analysis of simulation data including the delta method, steady-state estimation, and kernel density estimation
• Variance reduction, including importance sampling, latin hypercube sampling, and conditional Monte Carlo
• Estimation of derivatives and sensitivity analysis
• Advanced topics including cross-entropy, rare events, kernel density estimation, quasi Monte Carlo, particle systems, and randomized optimization

The presented theoretical concepts are illustrated with worked examples that use MATLAB®, a related Web site houses the MATLAB® code, allowing readers to work hands-on with the material and also features the author's own lecture notes on Monte Carlo methods. Detailed appendices provide background material on probability theory, stochastic processes, and mathematical statistics as well as the key optimization concepts and techniques that are relevant to Monte Carlo simulation.

Handbook of Monte Carlo Methods is an excellent reference for applied statisticians and practitioners working in the fields of engineering and finance who use or would like to learn how to use Monte Carlo in their research. It is also a suitable supplement for courses on Monte Carlo methods and computational statistics at the upper-undergraduate and graduate levels.

Preface.

Acknowledgments.

1 Uniform Random Number Generation.

1.1 Random Numbers.

1.2 Generators Based on Linear Recurrences.

1.3 Combined Generators.

1.4 Other Gnerators.

1.5 Tests for Random Number Generators.

References.

2 Quasirandom Number Generation.

2.1 Multidimensional Integration.

2.2 Van der Corput and Digital Sequences.

2.3 Halton Sequences.

2.4 Faure Sequences.

2.5 Sobol’ Sequences.

2.6 Lattice Methods.

2.7 Randomization and Scrambling.

References.

3 Random Variable Generation.

3.1 Generic Algorithms Based on Common Transformations.

3.2 Copulas.

3.3 Generation Methods for Various Random Objects.

References.

4 Probability Distributions.

4.1 Discrete Distributions.

4.2 Continuous Distributions.

4.3 Multivariate Distributions.

References.

5 Random Process Generation.

5.1 Gaussian Processes.

5.2 Markov Chains.

5.3 Markov Jump Processes.

5.4 Poisson Processes.

5.5 Wiener Process and Brownian Motion.

5.6 Stochastic Differential Equations and Diffusion Processes.

5.7 Brownian Bridge.

5.8 Geometric Brownian Motion.

5.9 Ornstein-Uhlenbeck Process.

5.10 Reflected Brownian Motion.

5.11 Fractional Brownian Motion.

5.12 Random Fields.

5.13 Lévy Processes.

5.14 Time Series.

References.

6 Markov Chain Monte Carlo.

6.1 Metropolis-Hastings Algorithm.

6.2 Gibbs Sampler.

6.3 Specialized Samplers.

6.4 Implementation Issues.

6.5 Perfect Sampling.

References.

7 Discrete Event Simulation.

7.1 Simulation Models.

7.2 Discrete Event Systems.

7.3 Event-Oriented Approach.

7.4 More Examples of Discrete Event Simulation.

References.

8 Statistical Analysis of Simulation Data.

8.1 Simulation Data.

8.2 Estimation of Performance Measures for Independent Data.

8.3 Estimation of Steady-State Performance Measures.

8.4 Emprical Cdf.

8.5 Kernal Density Estimation.

8.6 Resampling and the Bootstrap Method.

8.7 Goodness of Fit.

References.

9 Variance Reduction.

9.1 Variance Reduction Example.

9.2 Antithetic Random Variables.

9.3 Control Variables.

9.4 Conditional Monte Carlo.

9.5 Stratified Sampling.

9.6 Latin Hypercube Sampling.

9.7 Importance Scaling.

9.8 Quasi Monte Carlo

References.

10 Rare-Event Simulation.

10.1 Efficiency of Estimators.

10.2 Importance Sampling Methods for Light Tails.

10.3 Conditioning Methods for Heavy Tails.

10.4 State-Dependent Importance Sampling.

10.5 Cross-Entropy Method for Rare-Event Simulation.

10.6 Splitting Method.

References.

11 Estimation of Derivatives.

11.2 Finite Difference Method.

11.3 Infinitesimal Perturbation Analysis.

11.4 Score Function Method.

11.5 Weak Deriatives.

11.6 Sensitivity Analysis for Regenerative Processes.

References.

12 Randomized Optimization.

12.1 Stochastic Approximation.

12.2 Stochastic Counterpart Method.

12.3 Simulated Annealing.

12.4 Evolutionary Algorithms.

12.5 Cross-Entropy Method for Optimization.

12. 6 Other Randomized Optimization Techniques.

References.

13 Cross-Entropy Method.

13.1 Cross-Entropy Method.

13.2 Cross-Entropy Method for Estimation.

13.3 Cross-Entropy Method for Optimization.

References.

14 Particle Methods.

14.1 Sequential Monte Carlo.

14.2 Particle Splitting.

14.3 Splitting for Static Rare-Event Probability Estimaton.

14.5 Estimation of Multidimensional Integrals.

14.6 Combinatorial Optimization via Splitting.

14.7 Markov Chain Monte Carlo With Splitting.

References.

15 Applications to Finance.

15.1 Standard Model.

15.2 Pricing via Monte Carlo Simulation.

15.3 Sensitivities.

References.

16 Applications to Network  Reliability.

16.1 Network Reliability.

16.2 Evolution Model for a Static Network.

16.3 Conditional Monte Carlo.

16.4 Importance Sampling for Network Reliability.

16.5 Splitting Method.

References.

17 Applications to Differential Equations.

17. 1 Connections Between Stochastic and Partial Di_erential Equations.

17.2 Transport Processes and Equations.

17.3 Connections to ODEs Through Scaling.

References.

Appendix A: Probability and Stochastic Processes.

Appendix B: Elements of Mathematical Statistics.

Appendix C: Optimization.

Appendix D: Miscellany.

References.

Acronyms and Abbreviations.

List of Symbols.

List of Distributions.

Index.

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