### 2023　Stochastic differential equations

Font size  SML

Graduate major in Mathematical and Computing Science
Instructor(s)
Nakano Yumiharu  Miyoshi Naoto
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W9-322(W931))  Fri3-4(W9-322(W931))
Group
-
Course number
MCS.T419
Credits
2
2023
Offered quarter
4Q
Syllabus updated
2023/11/17
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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.

### Student learning outcomes

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 that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

Lecture-style

### Course schedule/Required learning

Course schedule Required learning
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.

### Out-of-Class Study Time (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.

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

Report

### Related courses

• MCS.T212 ： Fundamentals of Probability
• MCS.T312 ： Markov Analysis
• MCS.T410 ： Applied Probability

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

It is preferable that students have completed MCS.T212:Fundamentals of Probability and MCS.T312:Markov Analysis.