Random Processes in Physics and Finance

Random Processes in Physics and Finance

by Melvin Lax, Wei Cai, Min Xu

ISBN-10: 0198567766

ISBN-13: 9780198567769

Pub. Date: 11/30/2006

Publisher: Oxford University Press

This text is aimed at professionals and students working on random processes in various areas, including physics and finance. The first author, Melvin Lax (1922-2002) was a distinguished Professor of Physics as City College of New York and a member of the U.S. National Academy of Sciences, widely known for his contribution on random processes in physics. Most chapters


This text is aimed at professionals and students working on random processes in various areas, including physics and finance. The first author, Melvin Lax (1922-2002) was a distinguished Professor of Physics as City College of New York and a member of the U.S. National Academy of Sciences, widely known for his contribution on random processes in physics. Most chapters of this book are the outcome 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, starting from the basic probability theory, to Fokker-Planck and Langevin Processes, and includes diverse applications, such as explanation of very narrow laser width and analytical solution of the elastic Boltzmann transport equation. Lax's critical viewpoint on mathematics currently used in the financial world is also presented in this book.

About the Author:
Melvin Lax was a Distinguished Professor of Physics at the City College of New York

About the Author:
Wei Cai is part of the senior research staff at the Institute for Ultrafast Spectroscopy and Lasers, City College of New York

About the Author:
Min Xu is Assistant Professor of Physics at Fairfield University, Connecticut

Product Details

Oxford University Press
Publication date:
Oxford Finance Series
Product dimensions:
9.50(w) x 6.80(h) x 0.90(d)

Table of Contents

A Note from Co-authors     xiv
Review of probability     1
Meaning of probability     1
Distribution functions     4
Stochastic variables     5
Expectation values for single random variables     5
Characteristic functions and generating functions     7
Measures of dispersion     8
Joint events     12
Conditional probabilities and Bayes' theorem     16
Sums of random variables     19
Fitting of experimental observations     24
Multivariate normal distributions     29
The laws of gambling     32
Appendix A: The Dirac delta function     35
Appendix B: Solved problems     40
What is a random process     44
Multitime probability description     44
Conditional probabilities     44
Stationary, Gaussian and Markovian processes     45
The Chapman-Kolmogorov condition     46
Examples of Markovian processes     48
The Poisson process     48
The one dimensional random walk     50
Gambler's ruin     52
Diffusion processes and the Einstein relation     54
Brownian motion     56
Langevin theory of velocities in Brownian motion     57
Langevin theory of positions in Brownian motion     60
Chaos     64
Appendix A: Roots for the gambler's ruin problem     64
Appendix B: Gaussian random variables     66
Spectral measurement and correlation     69
Introduction: An approach to the spectrum of a stochastic process     69
The definitions of the noise spectrum     69
The Wiener-Khinchine theorem     71
Noise measurements     73
Evenness in [omega] of the noise?     75
Noise for nonstationary random variables     77
Appendix A: Complex variable notation     80
Thermal noise     82
Johnson noise     82
Equipartition     84
Thermodynamic derivation of Johnson noise     85
Nyquist's theorem     87
Nyquist noise and the Einstein relation     90
Frequency dependent diffusion constant     90
Shot noise     93
Definition of shot noise     93
Campbell's two theorems     95
The spectrum of filtered shot noise     98
Transit time effects     101
Electromagnetic theory of shot noise      104
Space charge limiting diode     106
Rice's generalization of Campbell's theorems     109
The fluctuation-dissipation theorem     113
Summary of ideas and results     113
Density operator equations     117
The response function     119
Equilibrium theorems     121
Hermiticity and time reversal     122
Application to a harmonic oscillator     123
A reservoir of harmonic oscillators     126
Generalized Fokker-Planck equation     129
Objectives     129
Drift vectors and diffusion coefficients     131
Average motion of a general random variable     134
The generalized Fokker-Planck equation     137
Generation-recombination (birth and death) process     139
The characteristic function     143
Path integral average     146
Linear damping and homogeneous noise     149
The backward equation     152
Extension to many variables     153
Time reversal in the linear case     160
Doob's theorem     162
A historical note and summary (M. Lax)     163
Appendix A: A method of solution of first order PDEs      164
Langevin processes     168
Simplicity of Langevin methods     168
Proof of delta correlation for Markovian processes     169
Homogeneous noise with linear damping     171
Conditional correlations     173
Generalized characteristic functions     175
Generalized shot noise     177
Systems possessing inertia     180
Langevin treatment of the Fokker-Planck process     182
Drift velocity     182
An example with an exact solution     184
Langevin equation for a general random variable     186
Comparison with Ito's calculus lemma     188
Extending to the multiple dimensional case     189
Means of products of random variables and noise source     191
The rotating wave van del Pol oscillator (RWVP)     194
Why is the laser line-width so narrow?     194
An oscillator with purely resistive nonlinearities     195
The diffusion coefficient     197
The van der Pol oscillator scaled to canonical form     199
Phase fluctuations in a resistive oscillator     200
Amplitude fluctuations     205
Fokker-Planck equation for RWVP     207
Eigenfunctions of the Fokker-Planck operator     208
Noise in homogeneous semiconductors     211
Density of states and statistics of free carriers     211
Conductivity fluctuations     215
Thermodynamic treatment of carrier fluctuations     216
General theory of concentration fluctuations     218
Influence of drift and diffusion on modulation noise     222
Random walk of light in turbid media     227
Introduction     227
Microscopic statistics in the direction space     229
The generalized Poisson distribution p[subscript n](t)     232
Macroscopic statistics     233
Analytical solution of the elastic transport equation     237
Introduction     237
Derivation of cumulants to an arbitrarily high order     238
Gaussian approximation of the distribution function     242
Improving cumulant solution of the transport equation     245
Signal extraction in presence of smoothing and noise     258
How to deal with ill-posed problems     258
Solution concepts     259
Methods of solution     261
Well-posed stochastic extensions of ill-posed processes     264
Shaw's improvement of Franklin's algorithm      266
Statistical regularization     268
Image restoration     270
Stochastic methods in investment decision     271
Forward contracts     271
Futures contracts     272
A variety of futures     273
A model for stock prices     274
The Ito's stochastic differential equation     278
Value of a forward contract on a stock     281
Black-Scholes differential equation     282
Discussion     283
Summary     286
Spectral analysis of economic time series     288
Overview     288
The Wiener-Khinchine and Wold theorems     291
Means, correlations and the Karhunen-Loeve theorem     293
Slepian functions     295
The discrete prolate spheroidal sequence     298
Overview of Thomson's procedure     300
High resolution results     301
Adaptive weighting     302
Trend removal and seasonal adjustment     303
Appendix A: The sampling theorem     303
Bibliography     307
Index     323

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