Random Processes in Physics and Finance

by Lax, Melvin; Cai, Wei; Xu, Min

• ISBN13: 9780198567769
• ISBN10: 0198567766
• Format: Hardcover
• Copyright: 2006-11-30
• Publisher: Oxford University Press

Summary

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 Yor

Author Biography

Melvin Lax (Deceased)
Distinguished Professor of Physics
City College of New York
Melvin Lax was a Distinguished Professor of Physics at the City College of the City University of New York (1971-2002), and a member of the U. S. National Academy of Sciences (1983-2002).
He has been associated with Bell Laboratories as a member of the technical staff (1955-1972), as head of the Theoretical Physics Research Department (1962-1964) and as consultant to the Solid State Electronics Research Laboratory.
After receiving his PhD in Physics from MIT (1947), Dr. Lax advanced from assistant to full professor of Physics at Syracuse University, (1947-55). He has also taught at Princeton (Spring 1961) and at Oxford (1961-1962). Dr. lax has published more than 200 papers.
In 1999 Lax shared the Willis Lamb Medal for Laser Science and Quantum Optics.
Dr. Lax was listed Who's Who in America.
Wei Cai
Senior research staff
Institute for Ultrafast Spectroscopy and lasers
Department of Physics
City College of New York
Wei Cai received Ph. D degree in Physics from University of Houston in 1985. He also received a MS degree in computer science from City College of City University of New York in 1992. He joined the Department of Physics at the City College of the City University of New York as a research associate in 1985. Recently, he is a senior member of the research staff at the Institute for Ultrafast Spectroscopy and Lasers. His main research interests are in radiative transfer and optical image processing. He has published 55 papers and holds 4 U. S. Patent.
Min Xu
Research staff
Institute for Ultrafast Spectroscopy and lasers
Department of Physics
City College of New York
Min Xu received Ph. D degree in Physics from City University of New York in 2001. He is currently a research associate at the Institute for Ultrafast Spectroscopy and Lasers. He works at the interface of physics, engineering and biomedical sciences. His main research interests are in optical physics, stochastic processes and inverse problems in physical and biological sciences, in particular, biomedical optical spectroscopy and imaging. He has published 30 peer-reviewed journal papers and holds 1 U. S. Patent.

Table of Contents

A Note from Co-authors	xiv
Review of probability	1	(43)
Meaning of probability	1	(3)
Distribution functions	4	(1)
Stochastic variables	5	(1)
Expectation values for single random variables	5	(2)
Characteristic functions and generating functions	7	(1)
Measures of dispersion	8	(4)
Joint events	12	(4)
Conditional probabilities and Bayes' theorem	16	(3)
Sums of random variables	19	(5)
Fitting of experimental observations	24	(5)
Multivariate normal distributions	29	(3)
The laws of gambling	32	(3)
Appendix A: The Dirac delta function	35	(5)
Appendix B: Solved problems	40	(4)
What is a random process	44	(4)
Multitime probability description	44	(1)
Conditional probabilities	44	(1)
Stationary, Gaussian and Markovian processes	45	(1)
The Chapman--Kolmogorov condition	46	(2)
Examples of Markovian processes	48	(21)
The Poisson process	48	(2)
The one dimensional random walk	50	(2)
Gambler's ruin	52	(2)
Diffusion processes and the Einstein relation	54	(2)
Brownian motion	56	(1)
Langevin theory of velocities in Brownian motion	57	(3)
Langevin theory of positions in Brownian motion	60	(4)
Chaos	64	(1)
Appendix A: Roots for the gambler's ruin problem	64	(2)
Appendix B: Gaussian random variables	66	(3)
Spectral measurement and correlation	69	(13)
Introduction: An approach to the spectrum of a stochastic process	69	(1)
The definitions of the noise spectrum	69	(2)
The Wiener--Khinchine theorem	71	(2)
Noise measurements	73	(2)
Evenness in ω of the noise?	75	(2)
Noise for nonstationary random variables	77	(3)
Appendix A: Complex variable notation	80	(2)
Thermal noise	82	(11)
Johnson noise	82	(2)
Equipartition	84	(1)
Thermodynamic derivation of Johnson noise	85	(2)
Nyquist's theorem	87	(3)
Nyquist noise and the Einstein relation	90	(1)
Frequency dependent diffusion constant	90	(3)
Shot noise	93	(20)
Definition of shot noise	93	(2)
Campbell's two theorems	95	(3)
The spectrum of filtered shot noise	98	(3)
Transit time effects	101	(3)
Electromagnetic theory of shot noise	104	(2)
Space charge limiting diode	106	(3)
Rice's generalization of Campbell's theorems	109	(4)
The fluctuation--dissipation theorem	113	(16)
Summary of ideas and results	113	(4)
Density operator equations	117	(2)
The response function	119	(2)
Equilibrium theorems	121	(1)
Hermiticity and time reversal	122	(1)
Application to a harmonic oscillator	123	(3)
A reservoir of harmonic oscillators	126	(3)
Generalized Fokker--Planck equation	129	(39)
Objectives	129	(2)
Drift vectors and diffusion coefficients	131	(3)
Average motion of a general random variable	134	(3)
The generalized Fokker--Planck equation	137	(2)
Generation--recombination (birth and death) process	139	(4)
The characteristic function	143	(3)
Path integral average	146	(3)
Linear damping and homogeneous noise	149	(3)
The backward equation	152	(1)
Extension to many variables	153	(7)
Time reversal in the linear case	160	(2)
Doob's theorem	162	(1)
A historical note and summary (M. Lax)	163	(1)
Appendix A: A method of solution of first order PDEs	164	(4)
Langevin processes	168	(14)
Simplicity of Langevin methods	168	(1)
Proof of delta correlation for Markovian processes	169	(2)
Homogeneous noise with linear damping	171	(2)
Conditional correlations	173	(2)
Generalized characteristic functions	175	(2)
Generalized shot noise	177	(3)
Systems possessing inertia	180	(2)
Langevin treatment of the Fokker--Planck process	182	(12)
Drift velocity	182	(2)
An example with an exact solution	184	(2)
Langevin equation for a general random variable	186	(2)
Comparison with Ito's calculus lemma	188	(1)
Extending to the multiple dimensional case	189	(2)
Means of products of random variables and noise source	191	(3)
The rotating wave van del Pol oscillator (RWVP)	194	(17)
Why is the laser line-width so narrow?	194	(1)
An oscillator with purely resistive nonlinearities	195	(2)
The diffusion coefficient	197	(2)
The van der Pol oscillator scaled to canonical form	199	(1)
Phase fluctuations in a resistive oscillator	200	(5)
Amplitude fluctuations	205	(2)
Fokker--Planck equation for RWVP	207	(1)
Eigenfunctions of the Fokker--Planck operator	208	(3)
Noise in homogeneous semiconductors	211	(16)
Density of states and statistics of free carriers	211	(4)
Conductivity fluctuations	215	(1)
Thermodynamic treatment of carrier fluctuations	216	(2)
General theory of concentration fluctuations	218	(4)
Influence of drift and diffusion on modulation noise	222	(5)
Random walk of light in turbid media	227	(10)
Introduction	227	(2)
Microscopic statistics in the direction space	229	(3)
The generalized Poisson distribution pn(t)	232	(1)
Macroscopic statistics	233	(4)
Analytical solution of the elastic transport equation	237	(21)
Introduction	237	(1)
Derivation of cumulants to an arbitrarily high order	238	(4)
Gaussian approximation of the distribution function	242	(3)
Improving cumulant solution of the transport equation	245	(13)
Signal extraction in presence of smoothing and noise	258	(13)
How to deal with ill-posed problems	258	(1)
Solution concepts	259	(2)
Methods of solution	261	(3)
Well-posed stochastic extensions of ill-posed processes	264	(2)
Shaw's improvement of Franklin's algorithm	266	(2)
Statistical regularization	268	(2)
Image restoration	270	(1)
Stochastic methods in investment decision	271	(17)
Forward contracts	271	(1)
Futures contracts	272	(1)
A variety of futures	273	(1)
A model for stock prices	274	(4)
The Ito's stochastic differential equation	278	(3)
Value of a forward contract on a stock	281	(1)
Black--Scholes differential equation	282	(1)
Discussion	283	(3)
Summary	286	(2)
Spectral analysis of economic time series	288	(19)
Overview	288	(3)
The Wiener--Khinchine and Wold theorems	291	(2)
Means, correlations and the Karhunen--Loeve theorem	293	(2)
Slepian functions	295	(3)
The discrete prolate spheroidal sequence	298	(2)
Overview of Thomson's procedure	300	(1)
High resolution results	301	(1)
Adaptive weighting	302	(1)
Trend removal and seasonal adjustment	303	(1)
Appendix A: The sampling theorem	303	(4)
Bibliography	307	(16)
Index	323