Stochastic Systems : Theory and Applications

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  • Format: Hardcover
  • Copyright: 2002-01-01
  • Publisher: World Scientific Pub Co Inc
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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

Prefacep. v
Dynamical Systems and Their Characteristicsp. 1
Continuous Systemsp. 1
Linear Differential Systemsp. 23
Connections of Systemsp. 42
Stochastic Differential Systemsp. 55
Stochastic Discrete Systemsp. 59
Stochastic Composed Systems. Basic Problems of Stochastic Systems Theoryp. 80
Problemsp. 89
Random Variables, Processes and Functionsp. 97
Probability Spacesp. 97
Random Variablesp. 102
Conditional Probabilitiesp. 110
Probabilities in Finite Product of Spacesp. 114
Probabilities in Infinite Product of Spacesp. 125
Existence of Regular Conditional Probabilitiesp. 130
Probability Measure of Random Functionp. 133
Separable Random Functionsp. 149
Continuous Random Functionsp. 157
Differentiable Random Functionsp. 167
Problemsp. 176
Moments, Characteristic Functions, Functionals and Canonical Expansionsp. 179
Expectation of Random Functionp. 179
Expectation of Random Variable in Topological Linear Spacep. 187
Momentsp. 190
Conditional Momentsp. 209
Characteristic Functions and Functionalsp. 218
Sequences of Probability Measures and Characteristic Functionals. Distributions in Banach Spacesp. 230
Normal Distributions in Linear Spacesp. 246
Approximate Representation of Distributionsp. 266
Canonical Expansionsp. 284
Problemsp. 311
Stochastic Integrals, Spectral and Integral Canonical Representationsp. 319
Mean Square Limitsp. 319
Mean Square Continuity of Random Functionp. 326
Differentiation of Random Functionsp. 330
Integration of Random Function over Nonrandom Measurep. 336
Integration over Random Measurep. 343
Stochastic Integralp. 345
Generalized Random Functionsp. 366
Integral Canonical and Spectral Representationsp. 376
Problemsp. 405
General Theory of Stochastic Systems and Its Applicationsp. 425
Stochastic Differentialsp. 425
Stochastic Differential Equationsp. 442
Differential Equations Containing Random Functionsp. 453
Transformation of System Equations into Stochastic Equationsp. 459
Numerical Integration of Stochastic Differential Equationp. 463
Multi-Dimensional Distributions in Stochastic Differential Systemsp. 479
Multi-Dimensional Distributions in Stochastic Discrete and Continuous-Discrete Systemsp. 506
Distributions in Stochastic Composed Systemsp. 510
Distributions in Stochastic Infinite-Dimensional Systemsp. 518
Applications. Optimal Online Filteringp. 523
Problemsp. 543
Methods of Linear Stochastic Systems Theory and Their Applicationsp. 557
Spectral and Correlation Methodsp. 557
Methods of General Linear Stochastic Systems Theoryp. 575
Methods of Linear Theory of Stochastic Systems based on Canonical Expansionsp. 592
Methods of Linear of Stochastic Systems Theory based on Integral Canonical Representationsp. 613
Applications. Shaping Filters Design and Modelingp. 624
Applications. Optimal Online Linear Filteringp. 641
Problemsp. 649
Methods of Nonlinear Stochastic Systems Theory and Their Applicationsp. 661
Moments of State Vectorp. 661
Methods of Normal Approximation and Statistical Linearizationp. 673
Methods of General Nonlinear Stochastic Systems Theory. Parametrization of Multi-Dimensional Distributionsp. 691
Methods of Moments and Semiinvariantsp. 695
Methods based on Orthogonal Expansions and Quasimomentsp. 719
Structural Parametrization of Distributions. Ellipsoidal Approximation Methodp. 737
Methods based on Canonical Representationsp. 768
Applications. Analysis of Stochastic Systems and Analytical Modelingp. 784
Online Conditionally Optimal Nonlinear Fillteringp. 813
Problemsp. 825
Appendicesp. 837
Stochastic Ito Differentials of Typical Composite Functionsp. 837
Hermite Polynomials. Polynomials Orthogonal to Gamma and X[superscript 2]-Distibutionsp. 839
Statistical Linearization of Typical Composite Functionsp. 853
Integrands Evaluation in Equations of Methods based on Parametrization Distributionsp. 856
Basic Characteristics of Some Continuous Typical Linear Systemsp. 864
Structural Transformation Rulesp. 870
Some Definite Integrals and Special Functionsp. 876
Bibliographical Notesp. 879
Referencesp. 885
Notationsp. 899
Indexp. 901
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