Martingales and Brownian motion

MATH-330 / 5 crédits

Enseignant: Blanc Guillaume Marie

Langue: Anglais

Summary

Introduction to the theory of discrete-time martingales (optional stopping and convergence theorems) and its applications (gambler's ruin, branching processes, Pólya urns...). Introduction to Brownian motion.

Content

Martingales are stochastic processes with a very natural special property which enables probabilists to prove a lot of theorems. They play a fundamental role in Probability theory, as well as in its applications to other fields (e.g, finance, population modelling, statistics, ...).

The course will cover the following topics:

  • conditional expectation
  • discrete-time stochastic processes and martingales,
  • stopping times and optional stopping theorem,
  • convergence of martingales,
  • applications: gambler's ruin, branching processes, Pólya urns,
  • introduction to Brownian motion,
  • basic properties of Brownian motion.

Keywords

martingales, optional stopping theorem, martingale convergence theorem, applications of martingales, Brownian motion

Learning Prerequisites

Required courses

First and second year courses of the mathematics bachelor program

Recommended courses

Probability theory (MATH-432), Measures and integration (MATH-303)

Teaching methods

Blackboard lectures and exercise sessions

Expected student activities

Attending the lectures and working on the exercise sheets

Assessment methods

Written exam

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

Supervision

Office hours No
Assistants Yes
Forum Yes

Resources

Bibliography

"Measure Theory, Probability, and Stochastic Processes", Jean-François Le Gall, Springer (2022)

"Probability: Theory and Examples", Rick Durrett, Cambridge University Press (5th ed, 2019)

"Brownian motion", Peter Mörters and Yuval Peres, Cambridge University Press (2010)

"Brownian Motion, Martingales, and Stochastic Calculus", Jean-François Le Gall, Springer (2016)

Ressources en bibliothèque

Notes/Handbook

Some lecture notes will be made available during the semester

Moodle Link

Prerequisite for


 

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

Semaine de référence

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