Simulation and Monte Carlo

Glossary.

1 Introduction to simulation and Monte Carlo.

1.1 Evaluating a definite integral.

1.2 Monte Carlo is integral estimation.

1.3 An example.

1.4 A simulation using Maple.

1.5 Problems.

2 Uniform random numbers.

2.1 Linear congruential generators.

2.2 Theoretical tests for random numbers.

2.3 Shuffled generator.

2.4 Empirical tests.

2.5 Combinations of generators.

2.6 The seed(s) in a random number generator.

2.7 Problems.

3 General methods for generating random variates.

3.1 Inversion of the cumulative distribution function.

3.2 Envelope rejection.

3.3 Ratio of uniforms method.

3.4 Adaptive rejection sampling.

3.5 Problems.

4 Generation of variates from standard distributions.

4.1 Standard normal distribution.

4.2 Lognormal distribution.

4.3 Bivariate normal density.

4.4 Gamma distribution.

4.5 Beta distribution.

4.6 Chi-squared distribution.

4.7 Studentâs t distribution.

4.8 Generalized inverse Gaussian distribution.

4.9 Poisson distribution.

4.10 Binomial distribution.

4.11 Negative binomial distribution.

4.12 Problems.

5 Variance reduction.

5.1 Antithetic variates.

5.2 Importance sampling.

5.3 Stratified sampling.

5.4 Control variates.

5.5 Conditional Monte Carlo.

5.6 Problems.

6 Simulation and finance.

6.1 Brownian motion.

6.2 Asset price movements.

6.3 Pricing simple derivatives and options.

6.4 Asian options.

6.5 Basket options.

6.6 Stochastic volatility.

6.7 Problems.

7 Discrete event simulation.

7.1 Poisson process.

7.2 Time-dependent Poisson process.

7.3 Poisson processes in the plane.

7.4 Markov chains.

7.5 Regenerative analysis.

7.6 Simulating a G/G/1 queueing system using the three-phase method.

7.7 Simulating a hospital ward.

7.8 Problems.

8 Markov chain Monte Carlo.

8.1 Bayesian statistics.

8.2 Markov chains and the Metropolisâ"Hastings (MH) algorithm.

8.3 Reliability inference using an independence sampler.

8.4 Single component Metropolisâ"Hastings and Gibbs sampling.

8.5 Other aspects of Gibbs sampling.

8.6 Problems.

9 Solutions.

9.1 Solutions 1.

9.2 Solutions 2.

9.3 Solutions 3.

9.4 Solutions 4.

9.5 Solutions 5.

9.6 Solutions 6.

9.7 Solutions 7.

9.8 Solutions 8.

Appendix 1: Solutions to problems in Chapter 1.

Appendix 2: Random Number Generators.

Appendix 3: Computations of acceptance probabilities.

Appendix 4: Random variate generators (standard distributions).

Appendix 5: Variance Reduction.

Appendix 6: Simulation and Finance.

Appendix 7: Discrete event simulation.

Appendix 8: Markov chain Monte Carlo.

References.

Index.