The mathematics for control and filtering
"The Mathematics for Control and Filtering" is an advanced course that provides a comprehensive exploration of the mathematical foundations underlying control theory and filtering techniques. Beginning with the fundamentals of probability theory and stochastic processes, the course progresses through stochastic analysis to delve into the intricacies of filtering and stochastic control problems. The curriculum also addresses cutting-edge research topics, including filtering on manifolds and applications of neural networks in control systems.
Lecturer
Date
10th September ~ 17th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 13:30 - 16:55 | A3-4-101 | ZOOM B | 462 110 5973 | BIMSA |
Prerequisite
Probability, Ordinary Differential Equations and Some elementary background in analysis.
Syllabus
Lecture Schedule:
Lecture 1 (9.10) – Review of Probability Theory
Probability Spaces and Events
Elementary Properties
Random Variables and Expectation Values
Properties of the Expectation and Inequalities
Limits of Random Variables
Induced Measures, Independence, and Absolute Continuity
Lecture 2 (9.24) – Random Process
Elementary Properties of Conditional Expectation
Discrete Time Stochastic Processes and Filtrations
Martingales
Martingale Convergence Theorem
The Radon-Nikodym Theorem Revisited
Separable σ-algebras
Proof of the Radon-Nikodym Theorem
Conditional Expectations and Martingales
Kolmogorov Definition
Martingales, Supermartingales, Submartingales
Stopping Times and Optional Stopping
Supermartingale Inequality
Lecture 3 (10.8) – Continuous Process and the Wiener Process
Continuous Process
Continuous Time Stochastic Processes
Equivalent Processes and Measurability
Continuous Processes
Basic Properties and Uniqueness
Existence: A Multiscale Construction
Properties of the Wiener Process
White Noise
Lecture 4 (10.15) – The Ito Integral Part I
What is Wrong with the Stieltjes Integral?
Before Defining the Ito Integral
Revisiting the Stieltjes Integral
Ito Integral
A Bare-Bones Construction
The Full-Blown Ito Integral
Continuous Sample Paths
Localization
Elementary Properties
Lecture 5 (10.22) – The Ito Integral Part II
Ito Calculus
Girsanov's Theorem
Martingale Representation Theorem
Lecture 6 (10.29) – Stochastic Differential Equations
Stochastic Differential Equations: Existence and Uniqueness
Markov Property and Kolmogorov’s Equations
Weak and Strong Solutions
Wong-Zakai Approximation
Euler-Maruyama Method
Stochastic Stability
Beyond the Lipschitz Condition
Lecture 7 (11.5) – Introduction to Stochastic Control
Stochastic Control Problems and Dynamic Programming
Introduction to stochastic control theory
Dynamic programming and Bellman's equation in stochastic control
Controlled Stochastic Differential Equations
Definition and examples of controlled stochastic differential equations (SDEs)
Applications of controlled SDEs in finance and engineering
Numerical methods for solving controlled SDEs
Lecture 8 (11.12) – Filtering Problem
Introduction to Filtering Theory
Observing noisy data
The optimal filtering problem
Linear and non-linear filtering problems
Bayesian Framework for Filtering
Recursive Bayesian estimation
Kalman filter as a special case of linear filtering
Extensions to non-linear filtering
Lecture 9 (11.19) – The Finite Dimensional Filter
Finite Dimensional Filters
Conditions for finite-dimensional filters
Examples of finite-dimensional filtering problems
Relationship between Kalman filter and finite-dimensional filters
Applications of Finite Dimensional Filters
Practical examples in control and signal processing
Lecture 10 (11.26) – Numerical Filtering Algorithms
Overview of Numerical Methods for Filtering
Discrete-time approximations
Particle filters and sequential Monte Carlo methods
Comparison between particle filters and Kalman filters
Implementation Challenges
Computational efficiency
Dealing with high-dimensional state spaces
Strategies to improve performance of particle filters
Lecture 11 (12.3) – The Yau and Yau Algorithms
Introduction to Yau's Filtering Algorithm
Derivation of Yau's algorithm
Key properties and convergence results
Differences from traditional filtering techniques
Yau and Yau Algorithms in Practice
Real-world applications of Yau’s algorithm
Implementing Yau’s algorithm in high-dimensional settings
Lecture 12 (12.10) – The Feedback Particle Filter and Optimal Transport
Feedback Particle Filter (FPF)
Overview of FPF for nonlinear filtering
Connections to optimal transport theory
FPF algorithm and numerical implementation
Optimal Transport in Filtering
Using optimal transport methods to improve filter accuracy
Applications of optimal transport in particle filtering
Lecture 13 (12.17) – Optimal Stopping and Impulse Control
Optimal Stopping Problems
Definition of optimal stopping in stochastic control
Classical examples: American options, real options
Dynamic programming approach to optimal stopping
Impulse Control
Introduction to impulse control in stochastic systems
Applications in inventory control, maintenance, and finance
Solving impulse control problems using dynamic programming
Lecture 1 (9.10) – Review of Probability Theory
Probability Spaces and Events
Elementary Properties
Random Variables and Expectation Values
Properties of the Expectation and Inequalities
Limits of Random Variables
Induced Measures, Independence, and Absolute Continuity
Lecture 2 (9.24) – Random Process
Elementary Properties of Conditional Expectation
Discrete Time Stochastic Processes and Filtrations
Martingales
Martingale Convergence Theorem
The Radon-Nikodym Theorem Revisited
Separable σ-algebras
Proof of the Radon-Nikodym Theorem
Conditional Expectations and Martingales
Kolmogorov Definition
Martingales, Supermartingales, Submartingales
Stopping Times and Optional Stopping
Supermartingale Inequality
Lecture 3 (10.8) – Continuous Process and the Wiener Process
Continuous Process
Continuous Time Stochastic Processes
Equivalent Processes and Measurability
Continuous Processes
Basic Properties and Uniqueness
Existence: A Multiscale Construction
Properties of the Wiener Process
White Noise
Lecture 4 (10.15) – The Ito Integral Part I
What is Wrong with the Stieltjes Integral?
Before Defining the Ito Integral
Revisiting the Stieltjes Integral
Ito Integral
A Bare-Bones Construction
The Full-Blown Ito Integral
Continuous Sample Paths
Localization
Elementary Properties
Lecture 5 (10.22) – The Ito Integral Part II
Ito Calculus
Girsanov's Theorem
Martingale Representation Theorem
Lecture 6 (10.29) – Stochastic Differential Equations
Stochastic Differential Equations: Existence and Uniqueness
Markov Property and Kolmogorov’s Equations
Weak and Strong Solutions
Wong-Zakai Approximation
Euler-Maruyama Method
Stochastic Stability
Beyond the Lipschitz Condition
Lecture 7 (11.5) – Introduction to Stochastic Control
Stochastic Control Problems and Dynamic Programming
Introduction to stochastic control theory
Dynamic programming and Bellman's equation in stochastic control
Controlled Stochastic Differential Equations
Definition and examples of controlled stochastic differential equations (SDEs)
Applications of controlled SDEs in finance and engineering
Numerical methods for solving controlled SDEs
Lecture 8 (11.12) – Filtering Problem
Introduction to Filtering Theory
Observing noisy data
The optimal filtering problem
Linear and non-linear filtering problems
Bayesian Framework for Filtering
Recursive Bayesian estimation
Kalman filter as a special case of linear filtering
Extensions to non-linear filtering
Lecture 9 (11.19) – The Finite Dimensional Filter
Finite Dimensional Filters
Conditions for finite-dimensional filters
Examples of finite-dimensional filtering problems
Relationship between Kalman filter and finite-dimensional filters
Applications of Finite Dimensional Filters
Practical examples in control and signal processing
Lecture 10 (11.26) – Numerical Filtering Algorithms
Overview of Numerical Methods for Filtering
Discrete-time approximations
Particle filters and sequential Monte Carlo methods
Comparison between particle filters and Kalman filters
Implementation Challenges
Computational efficiency
Dealing with high-dimensional state spaces
Strategies to improve performance of particle filters
Lecture 11 (12.3) – The Yau and Yau Algorithms
Introduction to Yau's Filtering Algorithm
Derivation of Yau's algorithm
Key properties and convergence results
Differences from traditional filtering techniques
Yau and Yau Algorithms in Practice
Real-world applications of Yau’s algorithm
Implementing Yau’s algorithm in high-dimensional settings
Lecture 12 (12.10) – The Feedback Particle Filter and Optimal Transport
Feedback Particle Filter (FPF)
Overview of FPF for nonlinear filtering
Connections to optimal transport theory
FPF algorithm and numerical implementation
Optimal Transport in Filtering
Using optimal transport methods to improve filter accuracy
Applications of optimal transport in particle filtering
Lecture 13 (12.17) – Optimal Stopping and Impulse Control
Optimal Stopping Problems
Definition of optimal stopping in stochastic control
Classical examples: American options, real options
Dynamic programming approach to optimal stopping
Impulse Control
Introduction to impulse control in stochastic systems
Applications in inventory control, maintenance, and finance
Solving impulse control problems using dynamic programming
Reference
Fundamentals of Stochastic Filtering, Alan Bain and Dan Crisan
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English