Stochastic Systems : Theory and Applications

by Pugachev, V. S.; Sinitsyn, I. N.

• ISBN13: 9789810247423
• ISBN10: 9810247427
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: World Scientific Pub Co Inc

Summary

This book presents the general theory and methods of linear and nonlinear stochastic systems (StS), i.e. systems described by stochastic differential, integral, integrodifferential, difference etc equations in finite- and infinite-dimensional spaces. The StS theory is based on the equations for characteristic functions and functionals. The book outlines the StS structural theory, including direct numerical StS analysis, methods of parametrization of one- and multi-dimensional distributions, and canonical expansions. It also considers StS applications in mechanics and physics, biology and chemistry, control and information processing, operation research, and finance.

Table of Contents

Preface	p. v
Dynamical Systems and Their Characteristics	p. 1
Continuous Systems	p. 1
Linear Differential Systems	p. 23
Connections of Systems	p. 42
Stochastic Differential Systems	p. 55
Stochastic Discrete Systems	p. 59
Stochastic Composed Systems. Basic Problems of Stochastic Systems Theory	p. 80
Problems	p. 89
Random Variables, Processes and Functions	p. 97
Probability Spaces	p. 97
Random Variables	p. 102
Conditional Probabilities	p. 110
Probabilities in Finite Product of Spaces	p. 114
Probabilities in Infinite Product of Spaces	p. 125
Existence of Regular Conditional Probabilities	p. 130
Probability Measure of Random Function	p. 133
Separable Random Functions	p. 149
Continuous Random Functions	p. 157
Differentiable Random Functions	p. 167
Problems	p. 176
Moments, Characteristic Functions, Functionals and Canonical Expansions	p. 179
Expectation of Random Function	p. 179
Expectation of Random Variable in Topological Linear Space	p. 187
Moments	p. 190
Conditional Moments	p. 209
Characteristic Functions and Functionals	p. 218
Sequences of Probability Measures and Characteristic Functionals. Distributions in Banach Spaces	p. 230
Normal Distributions in Linear Spaces	p. 246
Approximate Representation of Distributions	p. 266
Canonical Expansions	p. 284
Problems	p. 311
Stochastic Integrals, Spectral and Integral Canonical Representations	p. 319
Mean Square Limits	p. 319
Mean Square Continuity of Random Function	p. 326
Differentiation of Random Functions	p. 330
Integration of Random Function over Nonrandom Measure	p. 336
Integration over Random Measure	p. 343
Stochastic Integral	p. 345
Generalized Random Functions	p. 366
Integral Canonical and Spectral Representations	p. 376
Problems	p. 405
General Theory of Stochastic Systems and Its Applications	p. 425
Stochastic Differentials	p. 425
Stochastic Differential Equations	p. 442
Differential Equations Containing Random Functions	p. 453
Transformation of System Equations into Stochastic Equations	p. 459
Numerical Integration of Stochastic Differential Equation	p. 463
Multi-Dimensional Distributions in Stochastic Differential Systems	p. 479
Multi-Dimensional Distributions in Stochastic Discrete and Continuous-Discrete Systems	p. 506
Distributions in Stochastic Composed Systems	p. 510
Distributions in Stochastic Infinite-Dimensional Systems	p. 518
Applications. Optimal Online Filtering	p. 523
Problems	p. 543
Methods of Linear Stochastic Systems Theory and Their Applications	p. 557
Spectral and Correlation Methods	p. 557
Methods of General Linear Stochastic Systems Theory	p. 575
Methods of Linear Theory of Stochastic Systems based on Canonical Expansions	p. 592
Methods of Linear of Stochastic Systems Theory based on Integral Canonical Representations	p. 613
Applications. Shaping Filters Design and Modeling	p. 624
Applications. Optimal Online Linear Filtering	p. 641
Problems	p. 649
Methods of Nonlinear Stochastic Systems Theory and Their Applications	p. 661
Moments of State Vector	p. 661
Methods of Normal Approximation and Statistical Linearization	p. 673
Methods of General Nonlinear Stochastic Systems Theory. Parametrization of Multi-Dimensional Distributions	p. 691
Methods of Moments and Semiinvariants	p. 695
Methods based on Orthogonal Expansions and Quasimoments	p. 719
Structural Parametrization of Distributions. Ellipsoidal Approximation Method	p. 737
Methods based on Canonical Representations	p. 768
Applications. Analysis of Stochastic Systems and Analytical Modeling	p. 784
Online Conditionally Optimal Nonlinear Filltering	p. 813
Problems	p. 825
Appendices	p. 837
Stochastic Ito Differentials of Typical Composite Functions	p. 837
Hermite Polynomials. Polynomials Orthogonal to Gamma and X[superscript 2]-Distibutions	p. 839
Statistical Linearization of Typical Composite Functions	p. 853
Integrands Evaluation in Equations of Methods based on Parametrization Distributions	p. 856
Basic Characteristics of Some Continuous Typical Linear Systems	p. 864
Structural Transformation Rules	p. 870
Some Definite Integrals and Special Functions	p. 876
Bibliographical Notes	p. 879
References	p. 885
Notations	p. 899
Index	p. 901
Table of Contents provided by Syndetics. All Rights Reserved.