Stochastic Evolution Systems
In this topic course, we will introduce 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. Applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes will then be further explored. Finally, chaos expansion for linear stochastic evolution systems will be briefly introduced. This topic course will appeal to anyone studying or working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.
Lecturer
Date
15th September ~ 1st December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 13:30 - 16:55 | Shuimo | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Stochastic Analysis, Partial Differential Equations
Syllabus
1. Introduction and Examples
2. Stochastic Integration in a Hilbert Space
3. Linear Stochastic Evolution Systems----Coercive Systems
4. Linear Stochastic Evolution Systems----Dissipative Systems
5. Ito’s Second-Order Parabolic Equations
6. Ito’s Second-Order Parabolic Equations in Weighted Sobolev Spaces
7. Ito’s Partial Differential Equations and Diffusion Processes
8. The Forward Filtering Equation
9. The Backward Filtering Equation, Interpolation and Extrapolation
10. Hypoellipticity of Ito’s Second Order Parabolic Equations
11. Fundamental Solution of the Filtering Equation
12. Chaos Expansion for Linear Stochastic Evolution Systems
2. Stochastic Integration in a Hilbert Space
3. Linear Stochastic Evolution Systems----Coercive Systems
4. Linear Stochastic Evolution Systems----Dissipative Systems
5. Ito’s Second-Order Parabolic Equations
6. Ito’s Second-Order Parabolic Equations in Weighted Sobolev Spaces
7. Ito’s Partial Differential Equations and Diffusion Processes
8. The Forward Filtering Equation
9. The Backward Filtering Equation, Interpolation and Extrapolation
10. Hypoellipticity of Ito’s Second Order Parabolic Equations
11. Fundamental Solution of the Filtering Equation
12. Chaos Expansion for Linear Stochastic Evolution Systems
Reference
[1] Rozovsky B L, & Lototsky S V. (2018). Stochastic evolution systems. 2nd edition, Springer Cham.
[2] Bain, A., & Crisan, D. (2009). Fundamentals of stochastic filtering. Springer.
[2] Bain, A., & Crisan, D. (2009). Fundamentals of stochastic filtering. Springer.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English