2018　Stochastic differential equations

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Graduate major in Mathematical and Computing Science
Instructor(s)
Nakano Yumiharu  Miyoshi Naoto
Course component(s)
Lecture
Day/Period(Room No.)
Tue3-4(W832)  Fri3-4(W832)
Group
-
Course number
MCS.T419
Credits
2
2018
Offered quarter
4Q
Syllabus updated
2018/3/20
Lecture notes updated
2019/2/19
Language used
English
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

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

Course schedule/Required learning

Course schedule Required learning
Class 1 Conditional expectation, Martingales Explain the definitions of conditional expectations and martingales, and prove its basis properties.
Class 2 Wiener processes Explain and prove basic properties of Wiener processes.
Class 3 Stochastic integration Explain how stochastic integration is constructed, and validate it.
Class 4 Stochastic differential equations Explain the definition, concept, and examples of stochastic differential equations.
Class 5 Stochastic differential equations Explain and prove basic properties of stochastic differential equations.
Class 6 Stochastic differential equations Explain and prove basic properties of stochastic differential equations.
Class 7 Estimation of stochastic differential equations: theory Explain estimation methods for stochastic differential equations.
Class 8 Optimal control of stochastic differential equations: theory Explain optimal control methods for stochastic differential equations.
Class 9 Function approximations Explain methods for function approximations.
Class 10 Viscosity solutions Explain the definition and basis properties of viscosity solutions.
Class 11 Numerical analysis of partial differential equations Explain and implement numerical methods for partial differential equations.
Class 12 Estimation of stochastic differential equations: computation Implement the estimation methods for stochastic differential equations.
Class 13 Optimal control of stochastic differential equations Implement the control methods for stochastic differential equations.
Class 14 Applications Explain applications.

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. 