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.
Probability space, discrete and continuous random variables; Conditional probability and conditional expectation; Generating functions, characteristic functions, limit theorems; Random walk and branching processes ; Convergence of random variables, modes of convergence ; Introduction of martingales, martingale convergence theorem, optional stopping.
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.
Lévy Processes are stochastic processes with stationary independent increments. They are often used to describe random phenomena with fluctuations involving jumps.
In this course, we will mainly introduce Lévy Processes with one-sided jumps and the associated fluctuation theory. The following topics will be covered: Lévy-Ito decomposition, subordinators, exponential martingale and Esscher transform, scale functions, solution to the exit problems, potential measures, Wiener-Hopf factorization, reflected Lévy processes and the associated excursion processes. If time allows, we will also briefly introduce applications of such Lévy processes in population models and in risk theory. There will be no exam for this course. Each student is expected to give a short presentation on related topics.
Emphasis is on the probabilistic aspects (stochastic processes) although some estimation (inference) questions will also be discussed.
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.