Duration
30h Th, 10h Pr, 20h Proj.
Number of credits
| Master in mathematics, research focus (Odd years, not organized in 2024-2025) | 8 crédits | |||
| Master in mathematics, teaching focus (Odd years, not organized in 2024-2025) | 8 crédits |
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the second semester
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
The study of stochastic processes and the associated stochastic analysis are among the most contemporain domains in mathematics. They are still on the rise. The objective of this course is to offer the student the necessary tools to enter a very active but demanding research field.
This course presents and defines stochastic processes as well as foundamental tools to deeply explore their properties. It devotes a particular interest for the studies of the Brownian motion and the martingales.
In Chapter 1, we present the basic notions of the course as well as firt examples of stochastic processes. We also focus on Kolmogorov existence theorem.
Chapter 2 deals with specific aspects of continuous-time stochastic processes. Among them, Kolmogorov continuity theorem gives a condition to guarantee the regularity of the trajectories.
In Chapter 3, after exploring some generalities about Gaussian processes, we define and characterize the Brownian motion. A construction of it, as a random series, is also presented. Finally, we list some related processes, together with some first properties.
On a different note, Chapter 4 is mainly devoted to the study of the martingales. We first introduce the notions of filtrations before focusing specifically on the martingales. The main objective of this chapter is to present the "Martinagle Stopping Theorem", Doob's inequalities and study the convergence of martingales, both in discret and continuous time.
Finally, in Chapter 5, we utilize various results met in the course in order to present more properties of the Brownian motion.
Learning outcomes of the learning unit
The objective is to offer the student the keys to understand and manipulate stochastic processes.
Prerequisite knowledge and skills
It is compulsory to have a solid background in mathematics (BA in mathematics).
The course "Introduction to Stochastic Processes" is not a pre-requisite.
Planned learning activities and teaching methods
Ex cathedra classes, exercice sessions and a personal work.
Mode of delivery (face to face, distance learning, hybrid learning)
Face-to-face course
Additional information:
The theory will be presented during "face-to-face" lectures.
Students will have to solve exercises, alone or by group, and these exercises will be corrected and discuted during lectures.
Course materials and recommended or required readings
Platform(s) used for course materials:
- eCampus
- MyULiège
Further information:
See French section
Exam(s) in session
Any session
- In-person
oral exam
Written work / report
Further information:
An oral examination consisting of theory and exercises will be organized. A personal work, which must be submitted one week before the exam, will also be requested.
Work placement(s)
Organisational remarks and main changes to the course
Course taught in French every two years
This course is organised every two years on a rolling basis with MATH0514-1 Stochastic analysis. Even it is interesting to take both courses (during the two years of the master) to get a larger knownledge in stochastic analysis, the content of both courses are thought in such a way that they are independent.
Contacts
Laurent Loosveldt
Institut de Mathématique - B37 - Bureau 0/59
Quartier Polytech 1
Allée de la découverte, 12
4000 Liège (Sart-Tilman)
Tél. : (04) 366.92.56.
E-mail : l.loosveldt@uliege.be