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