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