Random Processes in Physics and Finance

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一本关于随机过程的书籍,希望对读者有所帮助.

目录如下:

1 Review of probability 1
1.1 Meaning of probability 1
1.2 Distribution functions 4
1.3 Stochastic variables 5
1.4 Expectation values for single random variables 5
1.5 Characteristic functions and generating functions 7
1.6 Measures of dispersion 8
1.7 Joint events 12
1.8 Conditional probabilities and Bayes'theorem 16
1.9 Sums of random variables 19
1.10 Fitting of experimental observations 24
1.11 Multivariate normal distributions 29
1.12 The laws of gambling 32
1.13 Appendix A: The Dirac delta function 35
1.14 Appendix B: Solved problems 40
2 What is a random process 44
2.1 Multitime probability description 44
2.2 Conditional probabilities 44
2.3 Stationary, Gaussian and Markovian processes 45
2.4 The Chapman-Kolmogorov condition 46
3 Examples of Markovian processes 48
3.1 The Poisson process 48
3.2 The one dimensional random walk 50
3.3 Gambler's ruin 52
3.4 Diffusion processes and the Einstein relation 54
3.5 Brownian motion 56
3.6 Langevin theory of velocities in Brownian motion 57
3.7 Langevin theory of positions in Brownian motion 60
3.8 Chaos 64
3.9 Appendix A: Roots for the gambler's ruin problem 64
3.10 Appendix B: Gaussian random variables 66
x CONTENTS
4 Spectral measurement and correlation 69
4.1 Introduction: An approach to the spectrum of a stochastic
process 69
4.2 The definitions of the noise spectrum 69
4.3 The Wiener-Khinchine theorem 71
4.4 Noise measurements 73
4.5 Evenness in u> of the noise? 75
4.6 Noise for nonstationary random variables 77
4.7 Appendix A: Complex variable notation 80
5 Thermal noise 82
5.1 Johnson noise 82
5.2 Equipartition 84
5.3 Thermodynamic derivation of Johnson noise 85
5.4 Nyquist's theorem 87
5.5 Nyquist noise and the Einstein relation 90
5.6 Frequency dependent diffusion constant 90
6 Shot noise 93
6.1 Definition of shot noise 93
6.2 Campbell's two theorems 95
6.3 The spectrum of filtered shot noise 98
6.4 Transit time effects 101
6.5 Electromagnetic theory of shot noise 104
6.6 Space charge limiting diode 106
6.7 Rice's generalization of Campbell's theorems 109
7 The fluctuation-dissipation theorem 113
7.1 Summary of ideas and results 113
7.2 Density operator equations 117
7.3 The response function 119
7.4 Equilibrium theorems 121
7.5 Hermiticity and time reversal 122
7.6 Application to a harmonic oscillator 123
7.7 A reservoir of harmonic oscillators 126
8 Generalized Fokker-Planck equation 129
8.1 Objectives 129
8.2 Drift vectors and diffusion coefficients 131
8.3 Average motion of a general random variable 134
8.4 The generalized Fokker-Planck equation 137
8.5 Generation-recombination (birth and death) process 139
CONTENTS xi
8.6 The characteristic function 143
8.7 Path integral average 146
8.8 Linear damping and homogeneous noise 149
8.9 The backward equation 152
8.10 Extension to many variables 153
8.11 Time reversal in the linear case 160
8.12 Doob's theorem 162
8.13 A historical note and summary (M. Lax) 163
8.14 Appendix A: A method of solution of first order PDEs 164
9 Langevin processes 168
9.1 Simplicity of Langevin methods 168
9.2 Proof of delta correlation for Markovian processes 169
9.3 Homogeneous noise with linear damping 171
9.4 Conditional correlations 173
9.5 Generalized characteristic functions 175
9.6 Generalized shot noise 177
9.7 Systems possessing inertia 180
10 Langevin treatment of the Fokker-Planck process 182
10.1 Drift velocity 182
10.2 An example with an exact solution 184
10.3 Langevin equation for a general random variable 186
10.4 Comparison with Ito's calculus lemma 188
10.5 Extending to the multiple dimensional case 189
10.6 Means of products of random variables and noise source 191
11 The rotating wave van del Pol oscillator (RWVP) 194
11.1 Why is the laser line-width so narrow? 194
11.2 An oscillator with purely resistive nonlinearities 195
11.3 The diffusion coefficient 197
11.4 The van der Pol oscillator scaled to canonical form 199
11.5 Phase fluctuations in a resistive oscillator 200
11.6 Amplitude fluctuations
11.7 Fokker-Planck equation for RWVP 207
11.8 Eigenfunctions of the Fokker-Planck operator 208
12 Noise in homogeneous semiconductors 211
12.1 Density of states and statistics of free carriers 211
12.2 Conductivity fluctuations
12.3 Thermodynamic treatment of carrier fluctuations
12.4 General theory of concentration fluctuations
205
215
216
218
xii CONTENTS
12.5 Influence of drift and diffusion on modulation noise 222
13 Random walk of light in turbid media 227
13.1 Introduction 227
13.2 Microscopic statistics in the direction space 229
13.3 The generalized Poisson distribution pn(t) 232
13.4 Macroscopic statistics 233
14 Analytical solution of the elastic transport equation 237
14.1 Introduction 237
14.2 Derivation of cumulants to an arbitrarily high order 238
14.3 Gaussian approximation of the distribution function 242
14.4 Improving cumulant solution of the transport equation 245
15 Signal extraction in presence of smoothing and noise 258
15.1 How to deal with ill-posed problems 258
15.2 Solution concepts 259
15.3 Methods of solution 261
15.4 Well-posed stochastic extensions of ill-posed processes 264
15.5 Shaw's improvement of Franklin's algorithm 266
15.6 Statistical regularization 268
15.7 Ima"&gev restoration 270
16 Stochastic methods in investment decision 271
16.1 Forward contracts 271
16.2 Futures contracts 272
16.3 A variety of futures 273
16.4 A model for stock prices 274
16.5 The Ito's stochastic differential equation 278
16.6 Value of a forward contract on a stock 281
16.7 Black-Scholes differential equation 282
16.8 Discussion 283
16.9 Summary 286
17 Spectral analysis of economic time series 288
17.1 Overview 288
17.2 The Wiener-Khinchine and Wold theorems 291
17.3 Means, correlations and the Karhunen-Loeve theorem 293
17.4 Slepian functions 295
17.5 The discrete prolate spheroidal sequence 298
17.6 Overview of Thomson's procedure 300
17.7 High resolution results 301
CONTENTS xiii
17.8 Adaptive weighting 302
17.9 Trend removal and seasonal adjustment 303
17.10 Appendix A: The sampling theorem 303
Bibliography 307
Index 323

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