This respected high-level text is aimed at students and professionals working on random processes in various areas, including physics and finance. The first author, Melvin Lax (1922-2002) was a distinguished Professor of Physics at City College of New York and a member of the U. S. National Academy of Sciences, and is widely known for his contributions to our understanding of random processes in physics. Most chapters of this book are outcomes of the class notes which Lax taught at the City University of New York from 1985 to 2001. The material is unique as it presents the theoretical framework of Lax's treatment of random processes, from basic probability theory to Fokker-Planck and Langevin Processes, and includes diverse applications, such as explanations of very narrow laser width, analytical solutions of the elastic Boltzmann transport equation, and a critical viewpoint of mathematics currently used in the world of finance.
Author(s): Melvin Lax, Wei Cai, Min Xu
Series: Oxford Finance Series
Publisher: Oxford University Press, USA
Year: 2006
Language: English
Pages: 327
Contents......Page 10
A Note from Co-authors......Page 15
1.1 Meaning of probability......Page 16
1.2 Distribution functions......Page 19
1.4 Expectation values for single random variables......Page 20
1.5 Characteristic functions and generating functions......Page 22
1.6 Measures of dispersion......Page 23
1.7 Joint events......Page 27
1.8 Conditional probabilities and Bayes' theorem......Page 31
1.9 Sums of random variables......Page 34
1.10 Fitting of experimental observations......Page 39
1.11 Multivariate normal distributions......Page 44
1.12 The laws of gambling......Page 47
1.13 Appendix A: The Dirac delta function......Page 50
1.14 Appendix B: Solved problems......Page 55
2.2 Conditional probabilities......Page 59
2.3 Stationary, Gaussian and Markovian processes......Page 60
2.4 The Chapman–Kolmogorov condition......Page 61
3.1 The Poisson process......Page 63
3.2 The one dimensional random walk......Page 65
3.3 Gambler's ruin......Page 67
3.4 Diffusion processes and the Einstein relation......Page 69
3.5 Brownian motion......Page 71
3.6 Langevin theory of velocities in Brownian motion......Page 72
3.7 Langevin theory of positions in Brownian motion......Page 75
3.9 Appendix A: Roots for the gambler's ruin problem......Page 79
3.10 Appendix B: Gaussian random variables......Page 81
4.2 The definitions of the noise spectrum......Page 84
4.3 The Wiener–Khinchine theorem......Page 86
4.4 Noise measurements......Page 88
4.5 Evenness in ω of the noise?......Page 90
4.6 Noise for nonstationary random variables......Page 92
4.7 Appendix A: Complex variable notation......Page 95
5.1 Johnson noise......Page 97
5.2 Equipartition......Page 99
5.3 Thermodynamic derivation of Johnson noise......Page 100
5.4 Nyquist's theorem......Page 102
5.6 Frequency dependent diffusion constant......Page 105
6.1 Definition of shot noise......Page 108
6.2 Campbell's two theorems......Page 110
6.3 The spectrum of filtered shot noise......Page 113
6.4 Transit time effects......Page 116
6.5 Electromagnetic theory of shot noise......Page 119
6.6 Space charge limiting diode......Page 121
6.7 Rice's generalization of Campbell's theorems......Page 124
7.1 Summary of ideas and results......Page 128
7.2 Density operator equations......Page 132
7.3 The response function......Page 134
7.4 Equilibrium theorems......Page 136
7.5 Hermiticity and time reversal......Page 137
7.6 Application to a harmonic oscillator......Page 138
7.7 A reservoir of harmonic oscillators......Page 141
8.1 Objectives......Page 144
8.2 Drift vectors and diffusion coefficients......Page 146
8.3 Average motion of a general random variable......Page 149
8.4 The generalized Fokker–Planck equation......Page 152
8.5 Generation–recombination (birth and death) process......Page 154
8.6 The characteristic function......Page 158
8.7 Path integral average......Page 161
8.8 Linear damping and homogeneous noise......Page 164
8.9 The backward equation......Page 167
8.10 Extension to many variables......Page 168
8.11 Time reversal in the linear case......Page 175
8.12 Doob's theorem......Page 177
8.13 A historical note and summary (M. Lax)......Page 178
8.14 Appendix A: A method of solution of first order PDEs......Page 179
9.1 Simplicity of Langevin methods......Page 183
9.2 Proof of delta correlation for Markovian processes......Page 184
9.3 Homogeneous noise with linear damping......Page 186
9.4 Conditional correlations......Page 188
9.5 Generalized characteristic functions......Page 190
9.6 Generalized shot noise......Page 192
9.7 Systems possessing inertia......Page 195
10.1 Drift velocity......Page 197
10.2 An example with an exact solution......Page 199
10.3 Langevin equation for a general random variable......Page 201
10.4 Comparison with Ito's calculus lemma......Page 203
10.5 Extending to the multiple dimensional case......Page 204
10.6 Means of products of random variables and noise source......Page 206
11.1 Why is the laser line-width so narrow?......Page 209
11.2 An oscillator with purely resistive nonlinearities......Page 210
11.3 The diffusion coefficient......Page 212
11.4 The van der Pol oscillator scaled to canonical form......Page 214
11.5 Phase fluctuations in a resistive oscillator......Page 215
11.6 Amplitude fluctuations......Page 220
11.7 Fokker–Planck equation for RWVP......Page 222
11.8 Eigenfunctions of the Fokker–Planck operator......Page 223
12.1 Density of states and statistics of free carriers......Page 226
12.2 Conductivity fluctuations......Page 230
12.3 Thermodynamic treatment of carrier fluctuations......Page 231
12.4 General theory of concentration fluctuations......Page 233
12.5 Influence of drift and diffusion on modulation noise......Page 237
13.1 Introduction......Page 242
13.2 Microscopic statistics in the direction space......Page 244
13.3 The generalized Poisson distribution p[sub(n)](t)......Page 247
13.4 Macroscopic statistics......Page 248
14.1 Introduction......Page 252
14.2 Derivation of cumulants to an arbitrarily high order......Page 253
14.3 Gaussian approximation of the distribution function......Page 257
14.4 Improving cumulant solution of the transport equation......Page 260
15.1 How to deal with ill-posed problems......Page 273
15.2 Solution concepts......Page 274
15.3 Methods of solution......Page 276
15.4 Well-posed stochastic extensions of ill-posed processes......Page 279
15.5 Shaw's improvement of Franklin's algorithm......Page 281
15.6 Statistical regularization......Page 283
15.7 Image restoration......Page 285
16.1 Forward contracts......Page 286
16.2 Futures contracts......Page 287
16.3 A variety of futures......Page 288
16.4 A model for stock prices......Page 289
16.5 The Ito's stochastic differential equation......Page 293
16.6 Value of a forward contract on a stock......Page 296
16.7 Black–Scholes differential equation......Page 297
16.8 Discussion......Page 298
16.9 Summary......Page 301
17.1 Overview......Page 303
17.2 The Wiener–Khinchine and Wold theorems......Page 306
17.3 Means, correlations and the Karhunen–Loeve theorem......Page 308
17.4 Slepian functions......Page 310
17.5 The discrete prolate spheroidal sequence......Page 313
17.6 Overview of Thomson's procedure......Page 315
17.7 High resolution results......Page 316
17.8 Adaptive weighting......Page 317
17.10 Appendix A: The sampling theorem......Page 318
Bibliography......Page 322
D......Page 338
H......Page 339
P......Page 340
T......Page 341
Z......Page 342
