MATH-330 / 5 crédits

Enseignant: Duch Pawel

Langue: Anglais

## 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

## 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

## 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)

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Martingales and Brownian motion
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

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