Stochastic Evolution Systems: Linear Theory and Applications to Non-Linear Filtering (Probability Theory and Stochastic Modelling, 89)

✍ Scribed by Boris L. Rozovsky, Sergey V. Lototsky

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 340
• Edition: 2nd ed. 2018
• Category: Library

✦ Synopsis

This monograph, now in a thoroughly revised second edition, develops the theory of stochastic calculus in Hilbert spaces and applies the results to the study of generalized solutions of stochastic parabolic equations.

The emphasis lies on second-order stochastic parabolic equations and their connection to random dynamical systems. The authors further explore applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes. The new edition now also includes a chapter on chaos expansion for linear stochastic evolution systems.

This book will appeal to anyone working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.

✦ Table of Contents

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 Introduction
2.2 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 Form
4.3 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 Property
6.3 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 Expansion
8.4 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