Explorations in Monte Carlo Methods

• Record Nr.: UNINA9910865269003321
• Autore: Shonkwiler Ronald W
• Pubbl/distr/stampa: Cham : , : Springer, , 2024; ©2024
• ISBN: 9783031559648; 9783031559631
• Edizione: [2nd ed.]
• Descrizione fisica: 1 online resource (290 pages)
• Collana: Undergraduate Texts in Mathematics Series
• Altri autori (Persone): MendivilFranklin
• Disciplina: 518.282
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Preface to the Second Edition -- Preface to the First Edition -- Acknowledgments -- Contents -- Notations -- 1 Introduction to Monte Carlo Methods -- 1.1 How Can Random Numbers Solve Problems? -- 1.1.1 History of the Monte Carlo Method -- 1.1.2 Histogramming Simulation Results -- 1.1.3 Sample Paths -- 1.2 Some Basic Probability -- 1.2.1 Events and Random Variables -- 1.2.2 Discrete and Continuous Random Variables -- 1.2.3 The Probability Density Function -- 1.2.4 Expected Values -- 1.2.5 Conditional Probabilities -- 1.2.6 Bayes' Formula -- 1.2.7 Joint Probability Distributions -- 1.3 Random Number Generation -- 1.3.1 Requirements for a Random Number Generator (RNG) -- 1.3.2 Middle-Square and Other Middle-Digit Techniques -- 1.3.3 Linear Congruential Random Number Generators -- 1.4 Some Applications -- 2 Some Probability Distributions and Their Uses -- 2.1 CDF Inversion-Discrete Case Example: Bernoulli Trials -- 2.1.1 Two-Outcome CDF Inversion -- 2.1.2 Multiple-Outcome Distributions -- 2.2 Walker's Alias Method Example: Roulette Wheel Selection -- 2.3 Probability Simulation Example: The Binomial Distribution -- 2.3.1 Sampling from the Binomial -- 2.4 Another Simulation Example: The Poisson Distribution -- 2.4.1 Sampling from the Poisson Distribution by Simulation -- 2.5 CDF Inversion, Continuous Case Example: The Exponential Distribution -- 2.5.1 Inverting the CDF-The Canonical Method for the Exponential -- 2.5.2 ; Discrete Event Simulation -- 2.5.3 Transforming Random Variables, the Cauchy Distribution -- 2.6 The Central Limit Theorem and the Normal Distribution -- 2.6.1 Sampling from the Normal Distribution -- 2.6.2 Approximate Sampling via the Central Limit Theorem -- 2.6.3 Error Estimates for Monte Carlo Simulations -- 2.7 Gibrat's Law and the Lognormal Distribution -- 2.8 Rejection Sampling Example: The Beta Distribution.; 2.8.1 The Beta Distribution -- 2.8.2 Sampling from an Unbounded Beta Distribution -- 2.9 Composite Distributions: Sampling the Gamma Distribution -- 2.9.1 The Gamma Distribution -- 2.9.2 Sampling from upper G left parenthesis alpha comma 1 right parenthesisG(α,1) -- 2.10 Sampling from a Joint Distribution -- 3 Markov Chain Monte Carlo -- 3.1 Discrete Markov Chains -- 3.1.1 Random Walk on a Graph -- 3.1.2 Matrix Representation of a Chain -- 3.2 Markov Chain Monte Carlo Sampling- The Metropolis Algorithm -- 3.2.1 Some Examples -- 3.2.2 Why Does the Metropolis Algorithm Work? -- 3.3 MCMC Sampling and the Ergodic Theorem -- 3.4 Statistical Mechanics -- 3.5 Ising Model and the Metropolis Algorithm -- 3.6 The Metropolis-Hastings Algorithm -- 3.7 Counting -- 3.8 Some Applications of MCMC -- 3.8.1 Shuffling with Constraints -- 3.8.2 Coupling from the Past -- 4 Random Walks -- 4.1 1d Random Walk -- 4.2 Diffusion -- 4.3 Brownian Motion -- 4.4 Random Walk Applications I -- 4.4.1 Options Pricing in Finance -- 4.4.2 Self-Avoiding Walks -- 4.5 Gambler's Ruin -- 4.5.1 Gambling Schemes -- 4.6 Random Walk Applications II-Kelly's Criterion in Finance -- 4.6.1 The Simple Kelly Game -- 4.6.2 The Simple Game with Catastrophic Loss -- 4.6.3 Option Trading Application I -- 4.6.4 Option Trading Application II -- 4.7 Random Walks and Electrical Networks -- 4.7.1 Markov Chain Solution for Voltages -- 4.7.2 The Fundamental Matrix and Expected Hitting Times -- 4.8 The Kinetic Monte Carlo Method -- 5 Optimization by Monte Carlo Methods -- 5.1 Simulated Annealing -- 5.2 Application of SA to the Traveling Salesman Problem -- 5.3 Genetic Algorithms -- 5.4 An Application of GA to Function Maximization -- 5.5 An Application of GA to the Permanent Problem -- 6 More on Markov Chain Monte Carlo -- 6.1 Bayesian Inference -- 6.1.1 Pymc3 -- 6.2 Gibbs Sampling.; 6.3 Monte Carlo Integration: Quadrature -- 6.3.1 Variance Reduction -- 6.3.2 MCMC in Quadrature -- 6.4 Round-off Error -- Appendix A Generating Uniform Random Numbers -- A.1 Multiple Stored Value Random Number Generation -- A.1.1 Fibonacci Generators -- A.1.2 Finite Field RNG -- A.2 Mersenne Twister -- A.3 Testing for Non-randomness -- A.3.1 Chi-Square Test -- A.3.2 Kolmogorov-Smirnov Test -- Appendix B Perron-Frobenius Theorem -- B.1 Proof of Perron-Frobenius -- Appendix C Kelly Allocation for Correlated Investments -- C.1 Kelly Allocation for Correlated Investments -- C.2 Genetic Algorithm Code for the Kelly Problem -- Appendix D Donsker's Theorem -- D.1 Donsker's Theorem -- Appendix E Projects -- Appendix References -- -- Index -- Code Index.