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.
Tribus et variables aléatoires. Théorie de l'intégration: théorème de Lebesgue, espace Lp, théorème de Fubini. Construction de mesures, mesure de Radon. Indépendance. Conditionnement.
This course will introduce a range of random graph processes and of random processes on graphs. I intend to cover the following models and topics, time permitting.
Conditional probability and conditional expectation, generating functions. Branching processes and random walk. Markov chains:transition matrices, classification of states, ergodic theorem, examples. Birth and death processes, queueing theory.
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.
Ce cours est une introduction au calcul stochastique pour les applications en finance mathématique:
1. Rappels de théorie des probabilités
2. Mouvement brownien et martingales
3. Intégration stochastique par rapport au mouvement brownien
4. Applications de la Formule d’Itô et Théorèmes de Girsanov
5. Équations différentielles stochastiques et processus de diffusion
6. Si le temps le permet : Introduction à la finance mathématique et au modèle de Black-Scholes-Merton, tarification d’options vanilles et d’options exotiques