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Table of Contents
Intro; Preface to the Second English Edition; Preface to the First English Edition; Contents; Standard Notations; 1 Examples and Auxiliary Results; 1.1 Introduction; 1.2 Examples of Stochastic Evolution Systems; 1.2.2 The Filtering Equation; 1.2.3 The Krylov Equation (Backward Diffusion Equation); 1.2.4 The Helmholtz Parabolic Equation; 1.2.5 A Continuous Branching Model with Geographical Structure; 1.2.6 Equation of the Free Field; 1.3 Measurability and Integrability in Banach Spaces; 1.4 Martingales in R1; 1.5 Diffusion Processes; 2 Stochastic Integration in a Hilbert Space
2.1 Introduction2.2 Martingales and Local Martingales; 2.3 Stochastic Integral with Respect to a Square Integrable Martingale; 2.4 Stochastic Integral with Respect to a Local Martingale; 2.5 An Energy Equality in a Rigged Hilbert Space; 3 Linear Stochastic Evolution Systems in Hilbert Spaces; 3.1 Introduction; 3.2 Coercive Systems; 3.3 Dissipative Systems; 3.4 Uniqueness and the Markov Property; 3.5 The First Boundary Value Problem for Itô Partial Differential Equations; 4 Itô's Second-Order Parabolic Equations; 4.1 Introduction
4.2 The Cauchy Problem for Super-Parabolic Itô Equations in Divergence Form4.3 The Cauchy Problem for Second-Order Parabolic Itô Equations in Non-divergence Form; 4.4 The Forward and Backward Cauchy Problems in Weighted Sobolev Spaces; 5 Itô's Partial Differential Equations and Diffusion Processes; 5.1 Introduction; 5.2 The Method of Stochastic Characteristics; 5.3 Inverse Diffusion Processes, Variation of Parameters and the Liouville Equations; 5.4 Representation of Measure-Valued Solutions; 6 Filtering, Interpolation and Extrapolation of Diffusion Processes; 6.1 Introduction
6.2 The Bayes Formula and the Conditional Markov Property6.3 The Forward Filtering Equation; 6.4 The Backward Filtering Equation, Interpolation, and Extrapolation; 7 Hypoellipticity of Itô's Second Order Parabolic Equations; 7.1 Introduction; 7.2 Measure-Valued Solution and Hypoellipticity Under a Generalized Hörmander Condition; 7.3 The Filtering Transition Density and the Fundamental Solution of the Filtering Equation in Hypoelliptic and Superparabolic Cases; 8 Chaos Expansion for Linear Stochastic Evolution Systems; 8.1 Introduction; 8.2 The Propagator
8.3 Additional Regularity by Chaos Expansion8.4 Chaos Expansion and Filtering of Diffusion Processes; 8.5 An Infinite-Dimensional Example; Notes; Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6; Chapter 7; Chapter 8; References; Index
2.1 Introduction2.2 Martingales and Local Martingales; 2.3 Stochastic Integral with Respect to a Square Integrable Martingale; 2.4 Stochastic Integral with Respect to a Local Martingale; 2.5 An Energy Equality in a Rigged Hilbert Space; 3 Linear Stochastic Evolution Systems in Hilbert Spaces; 3.1 Introduction; 3.2 Coercive Systems; 3.3 Dissipative Systems; 3.4 Uniqueness and the Markov Property; 3.5 The First Boundary Value Problem for Itô Partial Differential Equations; 4 Itô's Second-Order Parabolic Equations; 4.1 Introduction
4.2 The Cauchy Problem for Super-Parabolic Itô Equations in Divergence Form4.3 The Cauchy Problem for Second-Order Parabolic Itô Equations in Non-divergence Form; 4.4 The Forward and Backward Cauchy Problems in Weighted Sobolev Spaces; 5 Itô's Partial Differential Equations and Diffusion Processes; 5.1 Introduction; 5.2 The Method of Stochastic Characteristics; 5.3 Inverse Diffusion Processes, Variation of Parameters and the Liouville Equations; 5.4 Representation of Measure-Valued Solutions; 6 Filtering, Interpolation and Extrapolation of Diffusion Processes; 6.1 Introduction
6.2 The Bayes Formula and the Conditional Markov Property6.3 The Forward Filtering Equation; 6.4 The Backward Filtering Equation, Interpolation, and Extrapolation; 7 Hypoellipticity of Itô's Second Order Parabolic Equations; 7.1 Introduction; 7.2 Measure-Valued Solution and Hypoellipticity Under a Generalized Hörmander Condition; 7.3 The Filtering Transition Density and the Fundamental Solution of the Filtering Equation in Hypoelliptic and Superparabolic Cases; 8 Chaos Expansion for Linear Stochastic Evolution Systems; 8.1 Introduction; 8.2 The Propagator
8.3 Additional Regularity by Chaos Expansion8.4 Chaos Expansion and Filtering of Diffusion Processes; 8.5 An Infinite-Dimensional Example; Notes; Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6; Chapter 7; Chapter 8; References; Index