Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The members of the ISM Probability Group are involved in research in a broad range of areas spanning theoretical and applied, continuous and discrete probability. A particular focus is on the development and analysis of probabilistic models for real-world phenomena from physics, biology, statistics and computer science. Some specific topics of interest are: statistical physics in a random environment, branching systems in biology, distances and random energy landscapes, data structure analysis using random trees, genetics and population biology.
Many members of the group are also members of the CRM Probability Lab.
Students interested in graduate study in any of the areas cited above are invited to apply for admission to the program. There are no formal prerequisites other than those required by the departments. The following guidelines should be followed, however, and courses selected in consultation with an advisor from the group.
Students in the program are expected to have mastered the subject matter of the undergraduate curriculum in probability theory. All students are required to take the basic courses: Real Analysis Measure Theory and Probability Theory. Students are then expected to take a number of more specialized courses.
This course covers most of the materials in the first seven chapters of Probability and Random Processes by Grimmett and Stirzaker. In particular, it covers topics such as generating and characteristic functions and their applications in random walk and branching process, different modes of convergence and an introduction of martingales.
Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.
Espace de probabilité, variables aléatoires, indépendance, espérance mathématique, modes de convergence, lois des grands nombres, théorème central limite, espérance conditionnelle et martingales. Introduction au mouvement brownien.
Processus de branchement : modèles de Wright-Fisher, de Moran. Modèles à une infinité d’allèles, de sites. Facteurs d’évolution: sélection, mutation, migration, recombinaison, apparentement. Reconstruction et inférence de réseaux génétiques.
Tribus et variables aléatoires. Théorie de l'intégration: théorème de Lebesgue, espace Lp, théorème de Fubini. Construction de mesures, mesure de Radon. Indépendance. Conditionnement.
In the first part of this course we cover some basic topics on Markov chains, optimal stopping problems for Markov chains and discrete time Martingales. The second part starts with an introduction of various exotic properties of Brownian motion. We then introduce stochastic integrals with respect to Brownian motion, Ito's formula together with Girsanov transform and Feyman-Kac formula.
Characteristic functions: elementary properties, inversion formula, uniqueness, convolution and continuity theorems. Weak convergence. Central limit theorem. Additional topic(s) chosen (at discretion of instructor) from: Martingale Theory; Brownian motion, stochastic calculus.
Mouvement brownien, intégrale stochastique, formule d’Itô, équations différentielles stochastiques, théorèmes de représentation, théorème de Girsanov. Formule de Black et Scholes.