Martingales and Brownian motion
Summary
Introduction to the theory of discrete-time martingales including, in particular, the convergence and stopping time theorems. Application to branching processes. Introduction to Brownian motion and study of its properties.
Content
Martingales play a fundamental role in probability theory. Originally introduced to model betting strategies, they represent sequences of random variables where the expectation of the next value, given all past values, is equal to the current value. Martingales have wide range of practical applications across various fields: finance, statistic, machine learning, population modeling, epidemiology, etc.
The course will cover the following topics:
- introduction to the theory of martingales,
- stopping time and stopping time theorem,
- convergence of martingales,
- application to branching processes,
- introduction to Brownian motion,
- basic properties of Brownian motion.
Keywords
martingales, stopping time theorem, convergence theorem, branching processes, Brownian motion
Learning Prerequisites
Required courses
First and second year courses of the mathematics bachelor program
Recommended courses
Familiarity with the material covered in the courses 'Probability theory' (MATH-432) and 'Measure and integration' (MATH-303) will be helpful.
Teaching methods
Blackboard lectures and exercise sessions
Expected student activities
Attending the lectures and solving the exercises
Assessment methods
Written exam
In the case of Art. 3 para. 5 of the Section Regulations, the teacher decides on the format of the exam and communicates it to the concerned students.
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | No |
Resources
Bibliography
S. Karlin, H.M. Taylor, 'A First Course in Stochastic Processes', Academic Press (1975)
D. Williams, 'Probability with martingales', Cambridge University Press (1991)
P. Mörters, Y. Peres, 'Brownian motion', Cambridge University Press (2010)
Moodle Link
Prerequisite for
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Martingales and Brownian motion
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
Reference week
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Légendes:
Lecture
Exercise, TP
Project, Lab, other