Fall 2025, Instructor
金融隨機控制專題 Special Topics on Stochastic Control in Finance (QF528000)
Course Description & Audience:
The course focuses on studying stochastic systems and optimal control theory. We aim to provide a rigorous introduction to optimal control theory, covering both deterministic and stochastic control techniques with applications in quantitative finance, operations research, and management science. Topics include the Pontryagin Maximum Principle (PMP), dynamic programming, Hamilton-Jacobi-Bellman (HJB) equations, and their applications in dynamic optimization and control problems.
The intended topics to cover are listed below:
Part I: Foundations of Dynamic Optimization & Control
Preliminaries: Differential Equations and Optimization Basics
Bellman’s Principle of Optimality: Dynamic Programming
Pontryagin’s Maximum Principle
Finance Applications: Optimal Financing and Portfolio Choice
Part II: Stochastic Systems and Control
Stochastic Processes
Stochastic Differential Equations
Stochastic Control with Financial Applications
Optimal Consumption-Investment
Discrete-Time Stochastic Control
Model Predictive Control (Optional)
Prerequisites: Students planning to take this course should be fairly familiar with (advanced) mathematical analysis and some numerical software, such as Matlab or Python. If they are in doubt about their background, they should consult with the instructor.
Time and Place: Lectures are at R5R6R7, Room 735 TSMC Building. Meetings are also possible at other times by appointment.
Textbooks & References: Students will receive significant handout material to support the lectures. Below are some recommended textbooks; as we progress, other research papers will be assigned as reading tasks.
S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Boston, MA, USA: Springer, 2000.
D. P. Bertsekas, Dynamic Programming and Optimal Control, Belmont, MA, USA: Athena Scientific, 2017.
M. I. Kamien and N. L. Schwartz, Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Amsterdam, Netherlands: Elsevier, 1991.
N. L. Stokey, R. E. Lucas Jr, E. C. Prescott, Recursive Methods in Economic Dynamics, Harvard University Press, 1989.
D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case. Belmont, MA, USA: Athena Scientific, 1996.
H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications. Berlin, Germany: Springer, 2009.
S. E. Shreve, Stochastic Calculus for Finance I & II, Springer, 2004.
J. Miao, Economic Dynamics in Discrete Time, MIT Press, 2020.
Teaching Method: Lecture, Discussion, and Student Presentation.
Homework: Approximately bi-weekly. (Optional)
Office Hours: By appointment.
Grading: The grade will be based on one midterm test (50%), and a special research project (50%). The instructor may exercise discretion up to 10% in each grading category.
Course Material
Handouts
Assignments
Exams (and the Topics Covered)
Midterm/Final Exam:
Toolbox
CVXPY package for Python (or CVX for MATLAB) will be used frequently when solving some convex optimization problems.
Overleaf: A user-friendly online LaTeX editor.
Writing Tips.
Writing Tips for Ph.D. Students, J. H. Cochrane, University of Chicago.
Final Project Policies.
Your final project should include the following ingredients.
Title
Ensure it is concise, informative, and reflective of the project's main findings.
Abstract
Provide a succinct summary of the problem, methodology, main results, and conclusions.
Introduction
Provide a literature survey explaining what has been done, then point out what is missing in the literature (For example, say there is some weakness of the existing approach, so you want to fix it by providing your approach. As another example, say there is an issue that no one has noticed, so you jump in. As the last example, you found a phenomenon from your experiment/simulation/dataset, so you formed your hypothesis and wish to test it). This should provide a good enough ground for motivating your work. Then provide what is the difference between your approach and others, and compare and contrast.
Problem Formulation or Preliminaries
If your work involves a theoretical setting, provide them in this section. Define every notion and the corresponding notations in a consistent and precise way. You should tell the reader exactly the main problem you want to tackle and the methodologies you adopted.
Main results or Empirical studies
The main claim or findings go first. Then, provide supporting evidence. For empirical studies, provide a detailed explanation about your data; e.g., where did you get the data? why did you choose this dataset, not others? what do you see from the result? Are there any good or bad findings? If you run a simulation, say how you conduct it precisely and concisely.
Concluding Remarks or/and Future Work
References
Your final project should be no more than 10 pages of A4 papers.
You are required to use LaTeX to write your project; Overleaf might be a handy editor.