Theory and Simulation of Random Phenomena : Mathematical Foundations and Physical Applications

245 Pages·2018·1.715 MB·English

by  Galli, Davide Emilio, Motta, Mario, Vitali, Ettore

UNITEXT for Physics
Ettore Vitali · Mario Motta 
Davide Emilio Galli
Theory and
Simulation
of Random
Phenomena
Mathematical Foundations and Physical
Applications
UNITEXT for Physics
Series editors
Paolo Biscari, Milano, Italy
Michele Cini, Roma, Italy
Attilio Ferrari, Torino, Italy
Stefano Forte, Milano, Italy
Morten Hjorth-Jensen, Oslo, Norway
Nicola Manini, Milano, Italy
Guido Montagna, Pavia, Italy
Oreste Nicrosini, Pavia, Italy
Luca Peliti, Napoli, Italy
Alberto Rotondi, Pavia, Italy
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More information about this series at http://www.springer.com/series/13351
Ettore Vitali Mario Motta
(cid:129)
Davide Emilio Galli
Theory and Simulation
of Random Phenomena
Mathematical Foundations and Physical
Applications
123
Ettore Vitali Davide EmilioGalli
Department ofPhysics Department ofPhysics
Collegeof William andMary University of Milan
Williamsburg Milan
USA Italy
Mario Motta
Division of Chemistry andChemical
Engineering
California Institute ofTechnology
Pasadena
USA
ISSN 2198-7882 ISSN 2198-7890 (electronic)
UNITEXTfor Physics
ISBN978-3-319-90514-3 ISBN978-3-319-90515-0 (eBook)
https://doi.org/10.1007/978-3-319-90515-0
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Contents
1 Review of Probability Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Probability Spaces and Random Variables . . . . . . . . . . . . . . . . . 1
1.2 First Examples: Binomial Law, Poisson Law,
and Geometric Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Probability and Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 A Bit of Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Absolutely Continuous Random Variables . . . . . . . . . . . . . . . . . 11
1.5 Integration of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Integration with Respect to the Law of a Random
Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Transformations Between Random Variables . . . . . . . . . . . . . . . 20
1.7 Multi-dimensional Random Variables. . . . . . . . . . . . . . . . . . . . . 22
1.7.1 Evaluation of Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.8 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8.1 Moments and Cumulants . . . . . . . . . . . . . . . . . . . . . . . . 28
1.9 Normal Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.10 Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . 32
1.11 The Law of Large Numbers and the Central Limit Theorem . . . . 34
1.12 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Applications to Mathematical Statistics. . . . . . . . . . . . . . . . . . . . . . . 41
2.1 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 The Empiric Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Cochran Theorem and Estimation of the Variance . . . . . . . . . . . 46
2.4.1 The Cochran Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Estimation of a Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Cramer-Rao Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Maximum Likelihood Estimators (MLE) . . . . . . . . . . . . . . . . . . 55
vii
viii Contents
2.8 Hypothesis Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8.1 Student Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8.2 Chi-Squared Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.8.3 Kolmogorov-Smirnov Test . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Estimators of Covariance and Correlation. . . . . . . . . . . . . . . . . . 66
2.10 Linear Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.11 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Conditional Probability and Conditional Expectation. . . . . . . . . . . . 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 An Elementary Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Computing Conditional Expectations from Probability
Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 Properties of Conditional Expectation . . . . . . . . . . . . . . . . . . . . 82
3.7 Conditional Expectation as Prediction . . . . . . . . . . . . . . . . . . . . 83
3.8 Linear Regression and Conditional Expectation . . . . . . . . . . . . . 84
3.9 Conditional Expectation, Measurability and Independence. . . . . . 85
3.10 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Markov Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Random Walk in d Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 An Exact Expression for the Law. . . . . . . . . . . . . . . . . . 91
4.2.2 Explicit Expression in One Dimension
and Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.3 The Asymptotic Behavior for n! þ1
and Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Recursive Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Transition Matrix and Initial Law . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Invariant Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6 Metropolis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 Sampling of Random Variables and Simulation . . . . . . . . . . . . . . . . 109
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Random Number Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4 Simulation of Normal Random Variables. . . . . . . . . . . . . . . . . . 113
5.5 The Inverse Cumulative Distribution Function . . . . . . . . . . . . . . 114
Contents ix
5.6 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.7 The Metropolis Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.7.1 Monte Carlo Simulation of the Ising Model . . . . . . . . . . 116
5.7.2 Monte Carlo Simulation of a Classical Simple
Liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.8 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Brownian Motion: A Heuristic Introduction . . . . . . . . . . . . . . . . 132
6.3 Stochastic Processes: Basic Definitions . . . . . . . . . . . . . . . . . . . 134
6.3.1 Finite-Dimensional Laws . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Construction of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 Transition Probability and Existence of the Brownian
Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6 The Martingale Property and the Markov Property . . . . . . . . . . . 142
6.7 Wiener Measure and Feynman Path Integral. . . . . . . . . . . . . . . . 143
6.8 Markov Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.9 Semigroup Associated to a Markovian Transition Function . . . . . 148
6.10 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.11 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Stochastic Calculus and Introduction to Stochastic
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 Integration of Processes with Respect to Time . . . . . . . . . . . . . . 156
7.3 The Itô Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3.1 Itô Integral Integral of Simple Processes . . . . . . . . . . . . . 158
7.3.2 First Extension of the Itô Integral . . . . . . . . . . . . . . . . . . 161
7.3.3 Second Extension of the Itô Integral . . . . . . . . . . . . . . . . 162
7.3.4 The Itô Integral as a Function of Time . . . . . . . . . . . . . . 163
7.3.5 The Wiener Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4 Stochastic Differential and Itô’s Lemma. . . . . . . . . . . . . . . . . . . 165
7.5 Extension to the Multidimensional Case. . . . . . . . . . . . . . . . . . . 167
7.6 The Langevin Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.6.1 Ohm’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.7 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
x Contents
8 Stochastic Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.1 General Introduction to Stochastic Differential Equations . . . . . . 177
8.2 Stochastic Differential Equations and Markov Processes . . . . . . . 180
8.2.1 The Chapman-Kolmogorov Equation . . . . . . . . . . . . . . . 181
8.3 Kolmogorov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3.1 The Fokker-Planck Equation. . . . . . . . . . . . . . . . . . . . . . 186
8.4 Important Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.4.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 188
8.4.2 Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.4.3 Langevin Equation in a Force Field . . . . . . . . . . . . . . . . 190
8.5 Feynman-Kac Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.6 Kakutani Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.7 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Solutions . .... .... .... .... ..... .... .... .... .... .... ..... .... 203