2024 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Stochastic differential equations
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Yumiharu Nakano / Naoto Miyoshi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue / 3-4 Fri
- Class
- -
- Course Code
- MCS.T419
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
Stochastic differential equations are fundamental tools for describing dynamics of irregularly varying functions, and are applied to many areas. This course aims to get students to learn the fundamental theory and computational methods of stochastic differential equations.
Course description and aims
By the end of this course, students will be able to model stochastic differential equations and apply them in various ways, and moreover to explain the validity, limitation, and development of the methods used there.
Keywords
Brownian motions, Martingales, Stochastic integration, Stochastic differential equations, diffusion Estimation of stochastic processes, Diffusion models
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Lecture-style
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Conditional expectation, Measurability, Martingales | Explain the definitions of conditional expectations, measurability, and martingales, and prove its basis properties. |
| Class 2 | Conditional expectation, Measurability, Martingales | Explain the definitions of conditional expectations, measurability, and martingales, and prove its basis properties. |
| Class 3 | Brownian motion | Explain and prove basic properties of Brownian motion. |
| Class 4 | Brownian motion | Explain and prove basic properties of Brownian motion. |
| Class 5 | Stochastic integration | Explain how stochastic integration is constructed, and validate it. |
| Class 6 | Stochastic integration | Explain how stochastic integration is constructed, and validate it. |
| Class 7 | Stochastic differential equations | Explain and prove basic properties of stochastic differential equations. |
| Class 8 | Stochastic differential equations | Explain and prove basic properties of stochastic differential equations. |
| Class 9 | Stochastic differential equations | Explain and prove basic properties of stochastic differential equations. |
| Class 10 | Stochastic differential equations | Explain and prove basic properties of stochastic differential equations. |
| Class 11 | Statistical inference | Explain statistical inference methods for stochastic differential equations. |
| Class 12 | Weak solutions | Explain the theory of weak solutions. |
| Class 13 | Reverse-time diffusion processes | Explain and prove basic properties of reverse-time diffusion processes |
| Class 14 | Diffusion models | Explain the diffusion models in machine learning. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No specific text
Reference books, course materials, etc.
Course materials can be found on T2SCHOLA.
Reference books:
1)B. Oksendal, Stochastic differential equations: an introduction with applications, Springer
2) W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, Springer
3) H. Pham, Continuous-time stochastic control and optimization with financial applications, Springer
Evaluation methods and criteria
Report
Related courses
- MCS.T212 ： Fundamentals of Probability
- MCS.T312 ： Markov Analysis
- MCS.T410 ： Applied Probability
Prerequisites
It is preferable that students have completed MCS.T212:Fundamentals of Probability and MCS.T312:Markov Analysis.