Graduate course: Lévy processes

Course given by: Andreas Kyprianou

Course content: This course will give an introduction to the theory of Lévy processes. Key themes that will appear in the course are Lévy-Itô decomposition, the Wiener-Hopf factorisation and fluctuation theory with special attention given to spectrally negative Lévy processes (processes with no positive jumps). Selected applications will be looked at from the theory of branching processes, insurance risk, queuing theory and optimal stopping. I will give lectures for at least the first third of the course and then I will ask willing attendees to present material in `seminar-style' for the remainder of the course. Here is a rough outline of the material I intend to cover

• Introduction and rough overview
• A detailed exposition of the Lévy-Itô decomposition
• Path variation
• Path regularity and the Pollaczek-Khintchine formula
• The Wiener-Hopf factorisation
• Potential analysis of Spectrally Negative Processes
• Application 1: optimal stopping/stochastic control (Ronnie)
• Application 2: continuous state branching processes (Matt)
• Application 3: a brief introduction to self similar processes (Juan Carlos)
• Application 4: skorohod embedding with LPs (Alex)

I have tried to get a better room but this will mean moving around the campus each week. I prefer to do this in order to get hold of a proper blackboard!

The course is based on the following graduate text book below.