Stochastic Evolution Systems: Linear Theory and Applications to Non-linear Filtering

1 Examples and Auxiliary Results.- 1.0. Introduction.- 1.1. Examples of Stochastic Evolution Systems.- 1.2. Measurability and Integrability in Banach Spaces.- 1.3. Martingales in ?1.- 1.4. Diffusion Processes.- 2 Stochastic Integration in a Hilbert Space.- 2.0. Introduction.- 2.1. Martingales and Local Martingales.- 2.2. Stochastic Integrals with Respect to Square Integrable Martingale.- 2.3. Stochastic Integrable with Respect to a Local Martingale.- 2.4. An Energy Equality in a Rigged Hilbert Space.- 3 Linear Stochastic Evolution Systems in Hilbert Spaces.- 3.0. Introduction.- 3.1. Coercive Systems.- 3.2. Dissipative Systems.- 3.3. Uniqueness and the Markov Property.- 3.4. The First Boundary Problem for Ito's Partial Differential Equations.- 4 Ito'S Second Order Parabolic Equations.- 4.0. Introduction.- 4.1. The Cauchy Problem for Superparabolic Ito's Second Order Parabolic Equations.- 4.2. The Cauchy Problem for Ito's Second Order Equations.- 4.3. The Forward Cauchy Problem and the Backward One in Weighted Sobolev Spaces.- 5 Ito's Partial Differential Equations and Diffusion Processes.- 5.0. Introduction.- 5.1. The Method of Stochastic Characteristics.- 5.2. Inverse Diffusion Processes, the Method of Variation of Constants and the Liouville Equations.- 5.3. A Representation of a Density-valued Solution.- 6 Filtering Interpolation and Extrapolation of Diffusion Processes.- 6.0. Introduction.- 6.1. Bayes' Formula and the Conditional Markov Property.- 6.2. The Forward Filtering Equation.- 6.3. The Backward Filtering Equation Interpolation and Extrapolation.- 7 Hypoellipticity of Ito's Second Order Parabolic Equations.- 7.0. Introduction.- 7.1. Measure-valued Solution and Hypoellipticity under Generalized Hörmander's Condition.- 7.2. The Filtering Transition Density and a Fundamental Solution of the Filtering Equation in Hypoelliptic and Superparabolic Cases.- Notes.- References.