Stochastic Evolution Systems

Inhaltsverzeichnis

• 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.