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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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Add Random Processes in Physics and Finance, 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, Random Processes in Physics and Finance to the inventory that you are selling on WonderClubX
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Add Random Processes in Physics and Finance, 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, Random Processes in Physics and Finance to your collection on WonderClub |