### 2021　Stochastic differential equations

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Graduate major in Mathematical and Computing Science
Instructor(s)
Nakano Yumiharu  Miyoshi Naoto
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue3-4()  Fri3-4()
Group
-
Course number
MCS.T419
Credits
2
2021
Offered quarter
4Q
Syllabus updated
2021/9/30
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

Stochastic differential equations are fundamental tools for describing dynamics of irregulaly varying functions, and are applied to many areas. This course aims to get students to learn the fundamental theory and computational methods for estimations and controls of stochastic differential equations.

### Student learning outcomes

By the end of this course, students will be able to bulid models and compute optimal controls of stochastic differential equations, and moreover to explain the validity, limitation, and development of the methods used there.

### Keywords

Martingales, Stochastic integration, Stochastic differential equations, diffusion processes, Estimation of stochastic processes, Control of stochastic processes, Hamilton-Jacobi-Bellman equations

### 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 Optimal control of stochastic differential equations: theory Explain optimal control methods for stochastic differential equations.
Class 12 Optimal control of stochastic differential equations Explain optimal control methods for stochastic differential equations.
Class 13 Function approximations Explain methods for function approximations.
Class 14 Numerical analysis of partial differential equations Explain and implement numerical methods for partial differential equations.

### 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 OCW-i.
Reference books:
1)B. Oksendal, Stochastic differential equaions: 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

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### 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. 