2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis H
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kazuhiro Yasuda
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (M-112(H117))
- Class
- -
- Course Code
- MTH.C504
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
In this course, we continue the introduction to Ito calculus (stochastic calculus) following “Advanced Analysis G.” The topics covered include Itô’s formula, stochastic differential equations (SDEs), and the Feynman–Kac theorem. Applications of Ito calculus to mathematical finance will also be discussed.
We begin by studying Ito’s formula, which plays a central role in computations within Ito calculus. We then examine the existence and uniqueness of solutions to stochastic differential equations under Lipschitz conditions. After that, we introduce the Feynman–Kac theorem, which provides a probabilistic representation of solutions to parabolic partial differential equations. Finally, as an application to mathematical finance, we study the pricing of European call options and the associated delta-hedging strategy.
Course description and aims
・ Be able to understand the idea behind the derivation of Itô’s formula and apply it in practice.
・ Be able to understand the existence and uniqueness of solutions to stochastic differential equations.
・ Be able to understand that the Feynman–Kac theorem provides a probabilistic representation of solutions to parabolic partial differential equations.
・ Be able to understand the fundamental theory underlying applications to mathematical finance.
・ Be able to understand the ideas behind the proofs throughout the course.
Keywords
Ito’s formula, stochastic differential equations, the Cameron–Martin–Maruyama–Girsanov theorem, the Feynman–Kac theorem, option pricing, and delta hedging
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Ito’s formula and illustrative applications |
Details will be provided each class session. |
| Class 2 | Existence and uniqueness of solutions to stochastic differential equations |
|
| Class 3 | Approximation methods for stochastic differential equations |
|
| Class 4 | The Cameron–Martin–Maruyama–Girsanov theorem |
|
| Class 5 | The Feynman–Kac theorem |
|
| Class 6 | Pricing of European call options (binomial model) |
|
| Class 7 | Pricing of European call options (Black–Scholes model) |
|
| Class 8 | Delta hedging |
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)
None in particular.
Reference books, course materials, etc.
Stochastic Differential Equations (H. Nagai, Kyoritsu) (in Japanese)
Stochastic Differential Equations (S. Taniguchi, Kyoritsu) (in Japanese)
Evaluation methods and criteria
Assignments (100%)
Related courses
- MTH.C361 ： Probability Theory
- MTH.C503 ： Advanced topics in Analysis G
Prerequisites
None in particular
Other
None in particular
Information about this lecture will be announced via T2SCHOLA.