Descriptif

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.

In both theories, we will be especially interested in the long time behaviour of the process.