After successful completion of the course, students are able to show excellence in special fields of probability.

After having successfully completed the course, students are able to work with random point processes, Levy processes and to apply them in various fields of mathematics.

A large part of these lectures will be devoted to Levy processes, that are continuous-time stochastic processes with independent and stationary increments, and play an important role in applications (e.g. financial and actuarial mathematics). Levy processes include the famous cases of the Poisson process and the Brownian motion, but also the so-called stable processes and subordinators. We will develop the theory of Levy processes starting from the basics. Some preliminary lectures will be devoted to random point processes and in particular to Poisson point processes, that play an important role in the construction of Levy processes.  Point processes are themselves a fascinating topic in probability, and in fact we will see a bit of the general theory (for instance, if time allows, some basic notions concerning determinantal point processes), beyond what is strictly needed for the lectures that follow.

Lectures will take place in English. They will take place in person (with no streaming or recording); only in case of new lockdown they will be online.

I will use parts of the following books (more precise literature will be given during the lectures)

- A. Kyprianou, Introductory lectures on fluctuations of Levy processes with applications

- J. Bertoin, Levy processes

 Office hours for students: Tuesdays 2 p.m. - 3 p.m.

oral exam

Measure  & Probability Theory. Preferably also some knowledge of Stochastic processes