Brief presentation. The goal of this course is to present the two main modern tools in
probability and essential objects from a theoretical perspective as well as in applications:
martingales and Markov chains. Both pertain to the theory of stochastic processes in
discrete time, namely sequences of random variables which are not independent, but
rather in which the law at a given time depends on the past.
Martingale theory constitutes a fantastic tool that for example allows to describe the
law of the time and position of the first entry of such a process in a given subset as well
as to establish almost sure convergence as time tends to infinity. Markov chains appear
very naturally in the modelisation of various phenomena for it describes the evolution of
a stochastic process in which at a given time, the law of the next position in fact only
depends on the present position and not the whole past trajectory.
Prerequisites. This course is meant to be a second course on probability theory. Famili-
arity with basic measure theory and probability such as random variables, their law and
expectation, independence, Lp spaces, different notion of convergences, Law of Large
Numbers & Central Limit Theorem will be assumed. Some of these notions will be briefly
recapped and are covered in details in some of the references below if needed. Some
familiarity with Python for numerical applications can be useful.
Practical informations. The course consists of 2h of lectures and 2h of exercises ses-
sions per week, leaving plenty of time to work on your own using e.g. the references
below. Preparing the exercise in advance is necessary for the sessions to be useful. The
exercise sessions will also discuss simulations in Python.
The grade will be based on two written exams, one in the middle of the course and a
final one.