Traditionally, non-quantum physics has been concerned with deterministic equations where the dynamics of the system are completely determined by initial conditions. A century ago the discovery of Brownian motion showed that nature need not be deterministic. However, it is only recently that there has been broad interest in nondeterministic and even chaotic systems, not only in physics but in ecology and economics. On a short term basis, the stock market is nondeterministic and often chaotic. Despite its significance, there are few books available that introduce the reader to modern ideas in stochastic systems. This book provides an introduction to this increasingly important field and includes a number of interesting applications.
Author(s): Douglas Henderson, Peter Plaschko
Year: 2006
Language: English
Pages: 240
CONTENTS ......Page 12
Preface ......Page 8
Introduction ......Page 16
Glossary ......Page 22
1.1. Probability Theory ......Page 25
1.2. Averages ......Page 28
1.3. Stochastic Processes the Kolmogorov Criterion and Martingales ......Page 33
1.4. The Gaussian Distribution and Limit Theorems ......Page 38
1.4.1. The central limit theorem ......Page 40
1.5. Transformation of Stochastic Variables ......Page 41
1.6. The Markov Property ......Page 43
1.6.1. Stationary Markov processes ......Page 44
1.7. The Brownian Motion ......Page 45
1.8. Stochastic Integrals ......Page 52
1.9. The Ito Formula ......Page 62
Appendix: Poisson Processes ......Page 69
Exercises ......Page 73
2. Stochastic Differential Equations ......Page 79
2.1.1. Growth of populations ......Page 80
2.1.2. Stratonovich equations ......Page 82
2.1.3. The problem of Ornstein-Uhlenbeck and the Maxwell distribution ......Page 83
2.1.4. The reduction method ......Page 87
2.1.5. Verification of solutions ......Page 89
2.2. White and Colored Noise Spectra ......Page 91
2.3. The Stochastic Pendulum ......Page 94
2.3.1. Stochastic excitation ......Page 96
2.3.2. Stochastic damping (B = r = 0; a # 0) ......Page 97
2.4. The General Linear SDE ......Page 100
2.5. A Class of Nonlinear SDE ......Page 103
2.6. Existence and Uniqueness of Solutions ......Page 108
Exercises ......Page 111
3.1. The Master Equation ......Page 115
3.2. The Derivation of the Fokker-Planck Equation ......Page 119
3.3. The Relation Between the Fokker-Planck Equation and Ordinary SDE's ......Page 122
3.4. Solutions to the Fokker-Planck Equation ......Page 128
3.5. Lyapunov Exponents and Stability ......Page 131
3.6.1. First order SDE's ......Page 134
3.6.2. Higher order SDE's ......Page 136
Appendix A. Small Noise Intensities and the Influence of Randomness Limit Cycles ......Page 141
Appendix B.1 The method of Lyapunov functions ......Page 148
Appendix B.2 The method of linearization ......Page 152
Exercises ......Page 154
4.1. Stochastic Partial Differential Equations ......Page 159
4.2.1. A deterministic one-dimensional wave equation ......Page 165
4.2.2. Stochastic initial conditions ......Page 168
4.3.1. Introduction ......Page 171
4.3.2. Mathematical methods ......Page 172
4.3.3. Examples of exactly soluble problems ......Page 176
4.3.4. Probability laws and moments of the eigenvalues ......Page 180
4.4.1. Introduction ......Page 184
4.4.2. The Black-Scholes market ......Page 186
Exercises ......Page 188
5.1. Random Numbers Generators and Applications ......Page 191
5.1.1. Testing of random numbers ......Page 192
5.2. The Convergence of Stochastic Sequences ......Page 197
5.3. The Monte Carlo Integration ......Page 199
5.4. The Brownian Motion and Simple Algorithms for SDE's ......Page 203
5.5. The Ito-Taylor Expansion of the Solution of a 1D SDE ......Page 205
5.6. Modified 1D Milstein Schemes ......Page 211
5.7. The Ito-Taylor Expansion for N-dimensional SDE's ......Page 213
5.8. Higher Order Approximations ......Page 217
5.9. Strong and Weak Approximations and the Order of the Approximation ......Page 220
Exercises ......Page 225
References ......Page 229
Fortran Programs ......Page 235
Index ......Page 237
