[随机过程高清书籍]Stochastic Processes：From Physics to Finance

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收藏 2015-04-04

Stochastic Processes： From Physics to Finance

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1 A First Glimpse of Stochastic Processes . . . . . . . . . . . . . . . . 1
1.1 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Random Walk on a Line . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 From Binomial to Gaussian . . . . . . . . . . . . . . . . . 6
1.2.2 From Binomial to Poisson . . . . . . . . . . . . . . . . . . 11
1.2.3 Log–Normal Distribution . . . . . . . . . . . . . . . . . . 13
1.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 A Brief Survey of the Mathematics of Probability Theory . . . . . . 17
2.1 Some Basics of Probability Theory . . . . . . . . . . . . . . . . . 17
2.1.1 Probability Spaces and Random Variables . . . . . . . . . 18
2.1.2 Probability Theory and Logic . . . . . . . . . . . . . . . . 21
2.1.3 Equivalent Measures . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 Distribution Functions and Probability Densities . . . . . . 30
2.1.5 Statistical Independence and Conditional Probabilities . . . 31
2.1.6 Central Limit Theorem . . . . . . . . . . . . . . . . . . . 33
2.1.7 Extreme Value Distributions . . . . . . . . . . . . . . . . . 35
2.2 Stochastic Processes and Their Evolution Equations . . . . . . . . 38
2.2.1 Martingale Processes . . . . . . . . . . . . . . . . . . . . 41
2.2.2 Markov Processes . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Itô Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . 53
2.3.2 Stochastic Differential Equations and the Itô Formula . . . 57
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Diffusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 The Random Walk Revisited . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Polya Problem . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.2 Rayleigh-Pearson Walk . . . . . . . . . . . . . . . . . . . 69
3.1.3 Continuous-Time Random Walk . . . . . . . . . . . . . . 72
3.2 Free Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1 Velocity Process . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.2 Position Process . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Caldeira-Leggett Model . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.1 Definition of the Model . . . . . . . . . . . . . . . . . . . 85
3.3.2 Velocity Process and Generalized Langevin Equation . . . 86
3.4 On the Maximal Excursion of Brownian Motion . . . . . . . . . . 90
3.5 Brownian Motionina Potential:Kramers Problem . . . . . . . . . 92
3.5.1 First Passage Time for One-dimensional Fokker-Planck
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.2 Kramers Result . . . . . . . . . . . . . . . . . . . . . . . 97
3.6 A First Passage Problem for Unbounded Diffusion . . . . . . . . . 98
3.7 Kinetic Ising Models and Monte Carlo Simulations . . . . . . . . 101
3.7.1 Probabilistic Structure . . . . . . . . . . . . . . . . . . . . 102
3.7.2 Monte Carlo Kinetics . . . . . . . . . . . . . . . . . . . . 102
3.7.3 Mean-Field Kinetic Ising Model . . . . . . . . . . . . . . 105
3.8 Quantum Mechanics as a Diffusion Process . . . . . . . . . . . . . 110
3.8.1 Hydrodynamics of Brownian Motion . . . . . . . . . . . . 110
3.8.2 Conservative Diffusion Processes . . . . . . . . . . . . . . 114
3.8.3 Hypothesis of Universal Brownian Motion . . . . . . . . . 115
3.8.4 Tunnel Effect . . . . . . . . . . . . . . . . . . . . . . . . 118
3.8.5 Harmonic Oscillator and Quantum Fields . . . . . . . . . . 122
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4 Beyond the Central Limit Theorem: Lévy Distributions . . . . . . . 131
4.1 Back to Mathematics:StableDistributions . . . . . . . . . . . . . 132
4.2 The Weierstrass Random Walk . . . . . . . . . . . . . . . . . . . 136
4.2.1 Definition and Solution . . . . . . . . . . . . . . . . . . . 137
4.2.2 Superdiffusive Behavior . . . . . . . . . . . . . . . . . . . 143
4.2.3 Generalization to Higher Dimensions . . . . . . . . . . . . 147
4.3 Fractal-Time Random Walks . . . . . . . . . . . . . . . . . . . . 150
4.3.1 A Fractal-Time Poisson Process . . . . . . . . . . . . . . . 151
4.3.2 Subdiffusive Behavior . . . . . . . . . . . . . . . . . . . . 154
4.4 A Way to Avoid Diverging Variance: The Truncated Lévy Flight . 155
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5 Modeling the Financial Market . . . . . . . . . . . . . . . . . . . . . 163
5.1 Basic Notions Pertaining to Financial Markets . . . . . . . . . . . 164
5.2 Classical Option Pricing: The Black-Scholes Theory . . . . . . . . 173
5.2.1 The Black-Scholes Equation: Assumptions and Derivation . 174
5.2.2 The Black-Scholes Equation: Solution and Interpretation . 179
5.2.3 Risk-NeutralValuation . . . . . . . . . . . . . . . . . . . 184
5.2.4 Deviations from Black-Scholes: Implied Volatility . . . . . 189
5.3 Models Beyond Geometric Brownian Motion . . . . . . . . . . . . 191
5.3.1 Statistical Analysis of StockPrices . . . . . . . . . . . . . 192
5.3.2 The Volatility Smile: Precursor to Gaussian Behavior? . . . 205
5.3.3 Are Financial Time Series Stationary? . . . . . . . . . . . 209
5.3.4 Agent Based Modeling of Financial Markets . . . . . . . . 214
5.4 Towards a Model of Financial Crashes . . . . . . . . . . . . . . . 221
5.4.1 Some Empirical Properties . . . . . . . . . . . . . . . . . 222
5.4.2 A Market Model: From Self-organization to Criticality . . . 225
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Appendix A Stable Distributions Revisited . . . . . . . . . . . . . . . . 237
A.1 Testing for Domains of Attraction . . . . . . . . . . . . . . . . . . 237
A.2 Closed-Form Expressions and AsymptoticBehavior . . . . . . . . 239
Appendix B Hyperspherical Polar Coordinates . . . . . . . . . . . . . . 243
Appendix C TheWeierstrass RandomWalk Revisited . . . . . . . . . . 247
Appendix D The Exponentially Truncated Lévy Flight . . . . . . . . . . 253
Appendix E Put–Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . 259
Appendix F Geometric Brownian Motion . . . . . . . . . . . . . . . . . 261
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273