Instructor: Igor Kortchemski

Random phenomena are modelled using modern probability theory, defined in the 1930s by Kolmogorov using measure theory as a cornerstone. This course aims to provide a deep understanding of this theory. It is indeed an asset to forge intuition, to understand the objects involved and to mobilize them in an applied or theoretical framework.

This course is designed for an audience with a variety of interests: it may be of interest to students wishing to deepen their study of probability theory on the one hand, and on the other hand it may be of interest to students who intend to use it in business applications (a good understanding of probability theory is essential in order to be able to orient oneself in the world of applications and to innovate there). Each week is devoted to a theme of measure theory, with applications related to probability, involving discussions around exercises. The last session is devoted to oral presentations.

The evaluation is based on a 30-minute oral presentation by pairs of a research or overview article on a model, which is attended by all students. The objective is both individual (to learn to read a primary source and to present its content orally in English in a given time) and collective (to see a variety of models and applications in probability).

The course is delivered in english.

Lecture notes and exercise sheets

Moodle webpage of the course.

Feedback of the students on this course:

2023-2024

2022-2023

Progression

The teaching takes place on Tuesday's afternoons.

The course contains a "lecture" part and an "exercise" part. The "lecture" part will be organised in a "flipped classroom" way: at week n, lecture notes are given (covering the content of the theme of week n+1), together with an exercise. You are asked to read these lecture notes for week n+1 try to solve the exercise and upload your work on Moodle. At week n+1, the key elements of the lecture notes which were given at week n will be discussed, the exercise will be discussed, and then a tutorial session with new exercises will take place.

Provisional programme:

• Week 0 (September 24): Introduction, description of the course.
• Week 1 (October 1): Measures. (σ-fields, generated σ-field, mesures, monotone class theorem)
• Week 2 (October 8): Measurable functions. (measurable functions, product σ-field, approximation by simple functions, Doob-Dynkin lemma, contructing measures and Stieltjes measures) 
• October 15: no course (Forum Ecole)
• Week 3 (October 22): Integration. (integral with respect to a measure, monotone convergence, Radon-Nikodym theorem, Fubini theorem, convergence theorems)
• October 29: Holidays
• Week 4 (November 5): Independence (Independence of σ-fields and of random variables, transfer theorem, independence and integration, dummy function method, independence of infinite families, Kolmogorov 0-1 law, Borel-Cantelli lemmas) and recap on different notions of convergence for random variables.
• Week 5 (November 12): Conditioning. (conditional expectation, conditional law, Gaussian vectors)
• Week 6 (November 19): Brownian motion (1/2). (construction)
• November 26: no course (X-Forum)
• Week 7 (December 3): Brownian motion (2/2). (properties)
  
• Oral presentations December 10th and December 17th.

Evaluation

Evaluation is done through an oral presentation of a research paper or a significant application.

List of presentations