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Chapter 1 Introduction
1£®1 Preliminary work
1£®2 Overview of the book

Chapter 2 Filtering model and the Zakai equation
2£®1 Stochastic filtering
2£®1£®1 A general introduction
2£®1£®2 The filtering equations
2£®1£®3 Finite dimensional filters
2£®1£®4 Filtering from point process observations
2£®2 The filtering model
2£®2£®1 Unobserved state process
2£®2£®2 Observations
2£®2£®3 The objective
2£®3 The innovations processes
2£®4 The Zakai equations
2£®4£®1 A new measure
2£®4£®2 The unnormalized filtering equations

Chapter 3 Numerical methods
3£®1 A finite dimensional filter
3£®1£®1 Two special cases
3£®1£®2 A finite dimensional filter
3£®2 The finite-state Markov chain approximation
3£®2£®1 Approximating Markov chain
3£®2£®2 Filter
3£®2£®3 Numerical solution
3£®3 Particle methods

Chapter 4 Linear stochastic PDEs
4£®1 Semigroup approach
4£®1£®1 Semigroup
4£®1£®2 Linear SPDEs
4£®2 The variational approach to linear parabolic SPDEs
4£®2£®1 General setting
4£®2£®2 Basic results for linear parabolic SPDEs with Gaussiau noise

Chapter 5 Unnormalized conditional density
5£®1 Assumptions
5£®2 Main results
5£®3 Finding the unnormalized conditional density
5£®4 Proof of Theorem 5£®3
5£®4£®1 Some existence and uniqueness results on stochastic PDEs
5£®4£®2 The backward SPDEs
5£®4£®3 The unnormalized conditional density

Chapter 6 Convergence results
6£®1 Introduction
6£®2 The mild form of the Zakal equation
6£®3 The stochastic integral
6£®4 Continuity theorem
6£®4£®1 Continuity theorem
6£®4£®2 Proofs
6£®5 Galerkin approximation

Chapter 7 Galerkin approximation
7£®1 SDEs
7£®1£®1 Analytical solution
7£®1£®2 Numerical methods for SDEs
7£®2 Basis functions
7£®2£®1 Gaussiau series
7£®2£®2 Hermite expansion
7£®3 The adaptive Galerkin approximation
7£®3£®1 Introduction
7£®3£®2 The adaptive Galerkin approximation
7£®3£®3 The adaptive Galerkin approximation with Hermite polynomials
7£®4 Proofs
7£®5 Simulation studies
7£®5£®1 Tables
7£®5£®2 Figures
7£®5£®3 Summary

Appendix A Auxiliary calculations
A£®1 Preliminaries
A£®1£®1 Sobolev spaces
A£®1£®2 Some spaces of processes
A£®2 Some Hilbert spaces
Appendix B List of frequently used notations and symbols
Bibliography