2025 (Current Year) Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Topics on Mathematical and Computing Science OA
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Yuji Shinozaki
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - unknown
- Class
- -
- Course Code
- MCS.T424
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Jul 9, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course provides an introduction to the theory and applications of numerical analysis for stochastic differential equations (SDEs). We will begin with a general overview of SDEs and then discuss computational challenges specific to their numerical treatment, particularly in applications such as mathematical finance (e.g., derivatives pricing) and machine learning.
The course systematically covers both probabilistic methods—including discretization schemes for SDEs (e.g., Euler–Maruyama and Kusuoka methods) and Monte Carlo techniques—and PDE-based approaches such as finite difference schemes and tree-based algorithms. Advanced topics like branching algorithms and the recombination of measures will also be introduced, bridging theoretical underpinnings with practical implementation.
A key feature of this course is its multifaceted approach to SDE numerical analysis, combining mathematical rigor, intuitive interpretation, numerical stability, implementability, and broad applicability. In order to equip students with practical skills for research and professional applications of stochastic modeling, simplified numerical experiments using Python may be incorporated. Prior experience with measure theory or stochastic differential equations is recommended but not strictly required, as key concepts will be reviewed. However, students should have a strong interest in both mathematics and computational programming.
Course description and aims
To develop the ability to understand the numerical analysis of stochastic differential equations (SDEs), ranging from basic concepts to cutting-edge discussions.
Keywords
Stochastic differential equations, weak and strong approximations, probabilistic methods (Monte Carlo methods), finite difference methods, Malliavin calculus, and Kusuoka approximation.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
The course will be primarily lecture-based, but some sessions will also include numerical experiments and exercises using Python.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction (Oct 3, Fri, Periods 5–8) | An overview of numerical analysis for SDEs, its distinctive role in computational mathematics, and its connections to finance (especially derivatives modeling) and machine learning. |
| Class 2 | Foundations of Probability and SDEs (Oct 3, Fri, Periods 5–8) | Review of basic concepts in probability theory and stochastic differential equations. |
| Class 3 | Probabilistic Methods: Algorithmic Foundations (Oct 16, Thu, Periods 5–6, 9–10) | An introduction to basic algorithms for discretizing SDEs (Euler–Maruyama method, Kusuoka approximation), along with error analysis and implementation procedures. Python-based demonstrations may be included. |
| Class 4 | Probabilistic Methods: Theory of Discretization (Oct 16, Thu, Periods 5–6, 9–10) | Theoretical analysis of discretization errors using modern mathematical tools such as Malliavin calculus, and comparison with computational behavior. |
| Class 5 | Probabilistic Methods: Pseudorandom and Quasi-random Numbers (Oct 17, Fri, Periods 5–8) | Generation methods and error evaluation of pseudorandom and quasi-random sequences, with numerical implementation alongside previously introduced discretization schemes. |
| Class 6 | Probabilistic Methods: Research Trends in Quasi-Monte Carlo (Oct 17, Fri, Periods 5–8) | Introduction to recent developments and research directions in quasi-Monte Carlo methods. |
| Class 7 | PDE-Based Methods I: Feynman–Kac Formula and Finite Differences (Oct 24, Fri, Periods 5–8) | SDE numerics can be broadly divided into probabilistic and PDE-based approaches. Depending on the problem's structure (e.g., dimensionality), different methods apply. This session focuses on PDE-based methods using the Feynman–Kac formula and finite difference schemes. |
| Class 8 | PDE-Based Methods II: Efficient Tree-Based Simulation (Oct 24, Fri, Periods 5–8) | As an advanced topic, we will discuss the theory and applications of simulation methods based on tree structures, which offer both numerical efficiency and structure-preserving properties. |
| Class 9 | SDEs and Machine Learning I (Oct 31, Fri, Periods 5–8) | Discussion of key application areas of SDEs in machine learning, including diffusion models and deep learning. |
| Class 10 | SDEs and Machine Learning II (Oct 31, Fri, Periods 5–8) | Continued discussion of SDEs in machine learning applications, such as generative modeling and neural SDEs. |
| Class 11 | Advanced Topics I (Nov 14, Fri, Periods 5–8) | Based on student interest, we will explore cutting-edge topics such as Kusuoka approximations, efficient tree methods, quasi-random number generation, stochastic Volterra equations, and forward–backward SDEs. |
| Class 12 | Advanced Topics II (Nov 14, Fri, Periods 5–8) | Based on student interest, we will explore cutting-edge topics such as Kusuoka approximations, efficient tree methods, quasi-random number generation, stochastic Volterra equations, and forward–backward SDEs. |
| Class 13 | Advanced Topics III (Nov 20, Thu, Periods 5–6, 9–10) | Based on student interest, we will explore cutting-edge topics such as Kusuoka approximations, efficient tree methods, quasi-random number generation, stochastic Volterra equations, and forward–backward SDEs. |
| Class 14 | Advanced Topics IV (Nov 20, Thu, Periods 5–6, 9–10) | Based on student interest, we will explore cutting-edge topics such as Kusuoka approximations, efficient tree methods, quasi-random number generation, stochastic Volterra equations, and forward–backward SDEs. |
Study advice (preparation and review)
Textbook(s)
No textbook will be assigned.
Reference books, course materials, etc.
Materials will be provided during lectures.
Evaluation methods and criteria
Grading will be based on a report assignment (with some consideration possibly given to class participation).
Related courses
- MCS.T212 ： Fundamentals of Probability
- MCS.T332 ： Data Analysis
- MCS.T312 ： Markov Analysis
- MCS.T410 ： Applied Probability
- MCS.T333 ： Information Theory
- MCS.T403 ： Statistical Learning Theory
- MCS.T419 ： Stochastic differential equations
Prerequisites
Prior exposure to measure-theoretic probability and stochastic differential equations is desirable, but no advanced background knowledge is assumed, as the necessary material will be reviewed. However, participants are expected to have a genuine interest in both mathematics and in numerical computation or programming. Students are required to bring a laptop computer to the lectures.