Stochastic Modeling

✍ Scribed by Springer International Publishing; Lanchier, Nicolas

• Publisher: Springer
• Year: 2017
• Tongue: English
• Leaves: 305
• Series: Universitext
• Edition: 1st edition 2017
• Category: Library

✦ Synopsis

Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes.



The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler's ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright -Fisher model, Kingman's coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and Matlab(TM).

✦ Table of Contents

Preface......Page 7
Contents......Page 11
Part I Probability theory......Page 14
1 Basics of measure and probability theory......Page 15
1.1 Limitations of the Riemann integral......Page 16
1.2 Construction of the abstract integral......Page 17
1.3 Main properties of the integral......Page 27
1.4 Exercises......Page 30
2 Distribution and conditional expectation......Page 37
2.1 Radon–Nikodým theorem......Page 38
2.2 Induced measure and distribution......Page 39
2.3 Conditional expectation......Page 41
2.4 Exercises......Page 48
3 Limit theorems......Page 53
3.1 Levels of convergence......Page 54
3.2 Strong law of large numbers......Page 59
3.3 Central limit theorem......Page 63
3.4 Exercises......Page 67
Part II Stochastic processes......Page 69
4 Stochastic processes: general definition......Page 70
5 Martingales......Page 75
5.1 Optional stopping theorem......Page 77
5.2 Martingale convergence theorem......Page 84
5.3 Uniformly integrable and regular martingales......Page 88
5.4 Reverse martingales and the law of large numbers......Page 92
5.5 Doob's inequality and convergence in Lp......Page 94
5.6 Exercises......Page 97
6 Branching processes......Page 102
6.2 Connection with martingales......Page 103
6.3 Probability of survival......Page 105
6.4 Mean and variance of the number of individuals......Page 106
6.5 Exercises......Page 108
7 Discrete-time Markov chains......Page 109
7.1 Multi-step transition probabilities......Page 110
7.2 Classification of states......Page 112
7.3 Stationary distribution......Page 116
7.4 Number of visits......Page 119
7.5 Convergence to the stationary distribution......Page 124
7.6 Exercises......Page 132
8 Symmetric simple random walks......Page 137
8.1 Infinite lattices......Page 139
8.2 Electrical networks......Page 142
8.3 The infinite collision property......Page 143
8.4 Exercises......Page 145
9 Poisson point and Poisson processes......Page 148
9.1 Poisson point process and the Poisson distribution......Page 149
9.2 Poisson process and the exponential distribution......Page 151
9.3 Superposition and thinning......Page 156
9.4 The conditioning property......Page 161
9.5 Exercises......Page 163
10 Continuous-time Markov chains......Page 168
10.1 Definition and main assumptions......Page 169
10.2 Connection with discrete-time Markov chains......Page 171
10.3 Stationary distribution......Page 175
10.4 Limiting behavior......Page 180
10.5 Birth and death processes......Page 187
10.6 Exercises......Page 193
Part III Special models......Page 197
11 Logistic growth process......Page 198
11.1 Model description......Page 199
11.2 Long-term behavior......Page 200
11.3 Quasi-stationary distribution......Page 202
11.4 Time to extinction......Page 203
11.5 Exercises......Page 204
12 Wright–Fisher and Moran models......Page 207
12.1 The binomial distribution......Page 208
12.2 Probability of fixation......Page 209
12.3 Diffusion approximation and time to absorption......Page 213
12.4 Kingman's coalescent......Page 214
12.5 Moran model with selection......Page 219
12.6 Exercises......Page 220
13 Percolation models......Page 223
13.2 Monotonicity of the percolation probability......Page 225
13.3 The critical phenomenon......Page 227
13.4 Oriented site percolation in two dimensions......Page 232
13.5 Critical value and contour argument......Page 233
13.6 Exercises......Page 236
14 Interacting particle systems......Page 239
14.1 General framework......Page 240
14.2 Invasion: the contact process......Page 242
14.3 Competition: the voter model......Page 243
14.4 Graphical representation......Page 244
14.5 Numerical simulations in finite volume......Page 245
14.6 Existence on infinite graphs......Page 247
15 The contact process......Page 249
15.1 Monotonicity and attractiveness......Page 251
15.2 Self-duality......Page 252
15.3 The critical value......Page 255
15.4 Overview of the contact process......Page 258
15.5 Exercises......Page 260
16 The voter model......Page 263
16.1 Duality with coalescing random walks......Page 264
16.2 Clustering versus coexistence......Page 265
16.3 Overview of the voter model......Page 267
16.4 Exercises......Page 270
17 Numerical simulations in C and Matlab......Page 273
17.1 Classical models......Page 274
17.2 Spatially implicit models......Page 280
17.3 Percolation models......Page 288
17.4 Interacting particle systems......Page 292
References......Page 298
Index......Page 303