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Preface to the Second Edition
Introduction
1. Essentials of Probability Theory and Mathematical Statistics
1.1 Main Concepts of Probability Theory
1.2 Random Processes: Basic Notions
1.3 Markov Times
1.4 Brownian Motion Processes
1.5 Some Notions from Mathematical Statistics
2. Martingales and Related Processes: Discrete Time
2.1 Supermartingales and Submartingales on a Finite Time Interval
2.2 Submartingales on an Infinite Time Interval, and the Theorem of Convergence
2.3 Regular Martingales: Levy's Theorem
2.4 Invariance of the Supermartingale Property for Markov Times: Riesz and Doob Decompositions
3. Martingales and Related Processes: Continuous Time
3.1 Right Continuous Supermartingales
3.2 Basic Inequalities, the Theorem of Convergence, and Invari-ance of the Supermartingale Property for Markov Times
3.3 Doob-Meyer Decomposition for Supermartingales
3.4 Some Properties of Predictable Increasing Processes
4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations
4.1 The Wiener Process as a Square Integrable Martingale
4.2 Stochastic Integrals: It6 Processes
4.3 It6's Formula
4.4 Strong and Weak Solutions of Stochastic Differential Equations:
5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process
5.1 Doob-Meyer Decomposition for Square Integrable Martingales
5.2 Representation of Square Integrable Martingales
5.3 The Structure of Functionals of a Wiener Process
5.4 Stochastic Integrals over Square Integrable Martingales
5.5 Integral Representations of the Martingales which are Conditional Expectations and the Fubini Theorem for Stochastic Integrals
5.6 The Structure of Functionals of Processes of the Diffusion Type
6. Nonnegative Supermartingales and Martingales and the Girsanov Theorem
6.1 Nonnegative Supermartingales
6.2 Nonnegative Martingales
6.3 The Girshanov Theorem and its Generalization
7. Absolute Continuity of Measures corresponding to the It Processes and Processes of the Diffusion Type
7.1 The It5 Processes, and the Absolute Continuity of their Measures with respect to Wiener Measure
7.2 Processes of the Diffusion Type: the Absolute Continuity of their Measures with respect to Wiener Measure
7.3 The Structure of Processes whose Measure is Absolutely Continuous with Respect to Wiener Measure
7.4 Representation of the It6 Processes as Processes of the Diffusion Type, Innovation Processes, and the Struc ture of Functionals on the It5 Process
7.5 The Case of Gaussian Processes
7.6 The Absolute Continuity of Measures of the It5 Processes with respect to Measures Corresponding to Processes of the Diffusion Type
7.7 The Cameron-Martin Formula
7.8 The Cramer-Wolfowitz Inequality
7.9 An Abstract Version of the Bayes Formula
8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes
8.1 Filtering: the Main Theorem
8.2 Filtering: Proof of the Main Theorem
8.3 Filtering of Diffusion Markov Processes
8.4 Equations of Optimal Nonlinear Interpolation
8.5 Equations of Optimal Nonlinear Extrapolation
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9 Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States
10 Optimal Linear Nonstationary Filtering
Bibliography
Index
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