Advanced topics in Analysis G

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.C503

Number of credits

100

Course offered

2026

Offered quarter

1Q

Syllabus updated

Mar 5, 2026

Language

English

Syllabus

Course overview and goals

n this course, as an introduction to Ito calculus (stochastic calculus), we study the basics of Brownian motion, martingale processes, and the Ito integral. Although these concepts have a wide range of applications, the lectures are given with particular emphasis on applications to mathematical finance (concrete applications are treated in “Advanced Analysis H”).

In this course, we introduce Ito calculus (stochastic calculus) by studying Brownian motion, martingale processes, and the foundations of the Itô integral. Although these topics have a wide range of applications, the lectures will be presented with a particular emphasis on their use in mathematical finance (practical applications will be covered in “Advanced Analysis H”).
We begin with a brief review of the fundamentals of probability theory and then organize the concept of conditional expectation as an extension of these basics. Next, we study Brownian motion, the canonical example of a continuous-time stochastic process. We then examine martingale processes, which play a central role in Ito calculus, focusing on their definition and fundamental properties. After preparing the necessary tools such as quadratic variation, we proceed to the construction of the Ito integral (stochastic integral) and explore its key properties.

Course description and aims

・ Be able to understand the definition and fundamental properties of conditional expectation.
・ Be able to understand the behavior of Brownian motion and its key properties.
・ Be able to understand the concept of martingale processes and their fundamental properties.
・ Be able to understand the definition and main properties of the Itô integral (stochastic integral).
・ Be able to understand the ideas behind the proofs throughout the course.

Keywords

Brownian Motion, Conditional Expectationi, Continuous Martingale, Ito Integral (Stochastic Integral), Ito Calculus (Stochastic Calculus)

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

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

Based on reports. Details will be provided in the class.

Related courses

• MTH.C361 ： Probability Theory
• MTH.C504 ： Advanced topics in Analysis H

Prerequisites

None in particular

Other

None in particular
Information about this lecture will be announced via T2SCHOLA.