Stochastic Processes
Some stochastic models
Definition of a stochastic process
Distribution functions
Stationary Processes
Introduction
Moment functions
Stationary processes
Random phase and amplitude
Estimation of mean value and covariance function
Stationary processes and the non-stationary reality
Monte Carlo simulation from covariance function
The Poisson Process and Its Relatives
Introduction
The Poisson process
Stationary independent increments
The covariance intensity function
Spatial Poisson process
Inhomogeneous Poisson process
Monte Carlo simulation of Poisson processes
Spectral Representations
Introduction
Spectrum in continuous time
Spectrum in discrete time
Sampling and the aliasing effect
A few more remarks and difficulties
Monte Carlo simulation from spectrum
Gaussian Processes
Introduction
Gaussian processes
The Wiener process
Relatives of the Gaussian process
The Lévy process and shot noise process
Simulation of Gaussian process from spectrum
Linear Filters—General Theory
Introduction
Linear systems and linear filters
Continuity, differentiation, integration
White noise in continuous time
Cross-covariance and cross-spectrum
AR, MA, and ARMA Models
Introduction
Auto-regression and moving average
Estimation of AR parameters
Prediction in AR and ARMA models
A simple non-linear model—the GARCH process
Monte Carlo simulation of ARMA processes
Linear Filters—Applications
Introduction
Differential equations with random input
The envelope
Matched filter
Wiener filter
Kalman filter
An example from structural dynamics
Monte Carlo simulation in continuous time
Frequency Analysis and Spectral Estimation
Introduction
The periodogram
The discrete Fourier transform and the FFT
Bias reduction—data windowing
Reduction of variance
Appendix A: Some Probability and Statistics
Appendix B: Delta Functions and Stieltjes Integrals
Appendix C: Kolmogorov’s Existence Theorem
Appendix D: Covariance/Spectral Density Pairs
Appendix E: A Historical Background
References
Index
