Stochastic Systems: Theory and Applications > 수학/통계학

Table of Contents
Preface
Ch. 1 Dynamical Systems and Their Characteristics 1
1.1 Continuous Systems 1
1.2 Linear Differential Systems 23
1.3 Connections of Systems 42
1.4 Stochastic Differential Systems 55
1.5 Stochastic Discrete Systems 59
1.6 Stochastic Composed Systems. Basic Problems of Stochastic Systems Theory 80
Ch. 2 Random Variables, Processes and Functions 97
2.1 Probability Spaces 97
2.2 Random Variables 102
2.3 Conditional Probabilities 110
2.4 Probabilities in Finite Product of Spaces 114
2.5 Probabilities in Infinite Product of Spaces 125
2.6 Existence of Regular Conditional Probabilities 130
2.7 Probability Measure of Random Function 133
2.8 Separable Random Functions 149
2.9 Continuous Random Functions 157
2.10 Differentiable Random Functions 167
Ch. 3 Moments, Characteristic Functions, Functionals and Canonical Expansions 179
3.1 Expectation of Random Function 179
3.2 Expectation of Random Variable in Topological Linear Space 187
3.3 Moments 190
3.4 Conditional Moments 209
3.5 Characteristic Functions and Functionals 218
3.6 Sequences of Probability Measures and Characteristic Functionals. Distributions in Banach Spaces 230
3.7 Normal Distributions in Linear Spaces 246
3.8 Approximate Representation of Distributions 266
3.9 Canonical Expansions 284
Ch. 4 Stochastic Integrals, Spectral and Integral Canonical Representations 319
4.1 Mean Square Limits 319
4.2 Mean Square Continuity of Random Function 326
4.3 Differentiation of Random Functions 330
4.4 Integration of Random Function over Nonrandom Measure 336
4.5 Integration over Random Measure 343
4.6 Stochastic Integral 345
4.7 Generalized Random Functions 366
4.8 Integral Canonical and Spectral Representations 376
Ch. 5 General Theory of Stochastic Systems and Its Applications 425
5.1 Stochastic Differentials 425
5.2 Stochastic Differential Equations 442
5.3 Differential Equations Containing Random Functions 453
5.4 Transformation of System Equations into Stochastic Equations 459
5.5 Numerical Integration of Stochastic Differential Equation 463
5.6 Multi-Dimensional Distributions in Stochastic Differential Systems 479
5.7 Multi-Dimensional Distributions in Stochastic Discrete and Continuous-Discrete Systems 506
5.8 Distributions in Stochastic Composed Systems 510
5.9 Distributions in Stochastic Infinite-Dimensional Systems 518
5.10 Applications. Optimal Online Filtering 523
Ch. 6 Methods of Linear Stochastic Systems Theory and Their Applications 557
6.1 Spectral and Correlation Methods 557
6.2 Methods of General Linear Stochastic Systems Theory 575
6.3 Methods of Linear Theory of Stochastic Systems based on Canonical Expansions 592
6.4 Methods of Linear of Stochastic Systems Theory based on Integral Canonical Representations 613
6.5 Applications. Shaping Filters Design and Modeling 624
6.6 Applications. Optimal Online Linear Filtering 641
Ch. 7 Methods of Nonlinear Stochastic Systems Theory and Their Applications 661
7.1 Moments of State Vector 661
7.2 Methods of Normal Approximation and Statistical Linearization 673
7.3 Methods of General Nonlinear Stochastic Systems Theory. Parametrization of Multi-Dimensional Distributions 691
7.4 Methods of Moments and Semiinvariants 695
7.5 Methods based on Orthogonal Expansions and Quasimoments 719
7.6 Structural Parametrization of Distributions. Ellipsoidal Approximation Method 737
7.7 Methods based on Canonical Representations 768
7.8 Applications. Analysis of Stochastic Systems and Analytical Modeling 784
7.9 Online Conditionally Optimal Nonlinear Filtering 813
Appendices 837
Bibliographical Notes 879
References 885
Notations 899
Index 901