Statistical physics and economics. Concepts, tools and applications

Front cover......Page 1
Preface......Page 2
Contents......Page 6
1.1 What Is a Complex System?......Page 9
1.2 Determinism Versus Chaos......Page 12
1.3 The Probability Distribution......Page 15
1.4 The Liouville Equation......Page 17
1.5 Econophysics......Page 18
2.1.1 Probability Distribution of Relevant Quantities......Page 21
2.1.2 Measures of Central Tendency......Page 22
2.1.3 Measure of Fluctuations Around the Central Tendency......Page 23
2.1.4 Moments and Characteristic Functions......Page 24
2.1.5 Cumulants......Page 25
2.2.1 The Formal Solution of the Liouville Equation......Page 26
2.2.2 The Nakajima–Zwanzig Equation......Page 27
2.3.2 Joint Probability......Page 32
2.4 Markov Approximation......Page 34
2.5.1 Differential Chapman–Kolmogorov Equation......Page 36
2.5.2 Deterministic Processes......Page 41
2.5.4 The Master Equation......Page 42
2.6.1 Stationarity......Page 43
2.6.2 Correlation......Page 44
2.6.3 Spectra......Page 45
2.7.1 The Mori–Zwanzig Equation......Page 46
2.7.2 Separation of Timescales......Page 50
2.7.3 Wiener Process......Page 51
2.7.4 Stochastic Differential Equations......Page 52
2.7.5 Ito’s Formula and the Fokker–Planck Equation......Page 54
3.1.1 Finance and Financial Mathematics Versus Econophysics......Page 57
3.1.2 Scales in Financial Data......Page 59
3.1.3 Measurement of Price Fluctuations......Page 61
3.2.1 Probability Distributions......Page 63
3.2.2 Ergodicity in Financial Data......Page 64
3.2.3 Stationarity of Financial Markets......Page 65
3.2.4 Markov Approximation......Page 68
3.2.5 Taxonomy of Stocks......Page 69
3.3.1 The Central Limit Theorem......Page 75
3.3.2 Convergence Problems......Page 80
3.3.3 Fokker–Planck Equation for Financial Processes......Page 82
3.3.4 The Log-Normal Law......Page 87
3.4.1 The Escape Problem......Page 88
3.4.2 The Portfolio Problem......Page 90
3.4.3 Option Pricing Theory......Page 94
3.5.1 High-Frequency Observations......Page 103
3.5.2 Lévy Distributions......Page 105
3.5.3 Convergence to Stable L´evy Distributions......Page 108
3.5.4 Scaling Behavior......Page 109
3.5.5 Truncated L´evy Distributions......Page 114
3.6.1 Extreme Value Theory......Page 116
3.6.2 Partition Function Formalism......Page 123
3.6.3 The Cramér Theorem......Page 128
3.6.4 Extreme Fluctuations......Page 130
3.6.5 A Mechanism for Extreme Price Changes......Page 132
3.7.1 Time Correlation in Financial Data......Page 134
3.7.2 Ultrashort Timescales......Page 138
3.7.3 Autoregressive Processes......Page 151
3.7.4 Time-Reversible Symmetry......Page 160
4.1 The Task of Economics......Page 165
4.2.1 The Polypolist and Stability Analysis......Page 173
4.2.2 The Monopolist and Chaotic Behavior......Page 178
4.3.1 Thermodynamic Concepts......Page 182
4.3.2 Consumer Behavior......Page 184
4.3.3 Thermodynamics and Environmental Economics......Page 188
4.4.1 Models and Measurements......Page 189
4.4.2 Scaling Laws and Scale Invariance......Page 192
4.4.3 Economic Field Theories......Page 196
5.1 Models and Simulations......Page 203
5.2.1 Monte Carlo and Random Generators......Page 205
5.2.2 Dynamic Monte Carlo......Page 207
5.2.3 Quasi-Monte Carlo......Page 208
5.2.4 Reverse Monte Carlo......Page 211
5.3 Cellular Automata......Page 212
6.1 Regression and Autoregression......Page 215
6.2.1 Public Surveys and Decision-Making......Page 218
6.2.2 Bayesian Theory and Forecasting......Page 220
6.3.1 Introduction......Page 221
6.3.2 Spin Glasses and Neural Networks......Page 222
6.3.3 Topology of Neural Networks......Page 226
6.3.4 Training of Neural Networks......Page 229
6.3.5 Neural Networks for Forecasting......Page 231
References......Page 235
Index......Page 249