# Levy Processes

 Course higher education credits 7.5 Course is normally given Period 2. Course first taught 2012 Graduate school Mathematics Department Mathematical Sciences Course start 2012-10-22 Course end 2012-12-21 Contact information   Adam Andersson, adam.andersson@chalmers.se Course description   This fall we will have a study group on the subject "Lévy processes and stochastic calculus", following the book by David Applebaum with the same title. We will study sufficiently much of Lévy processes so that we can reach our main objective to define and build the theory of stochastic integration with respect to Lévy processes and treat stochastic differential equations (SDEs) driven by the same. After a quick reading of the book this theory seems quite interesting. Central in the theory is the concept of infinitely divisible measures and the Lévy-Khintchine formula. This formula tells that every infinitely divisible measure has a Fourier transform of the type $exp(\eta)$ where the function $\eta$ is the so called Lévy-symbol of the measure. A Lévy process $X_t$ is a stochastic process whose distribution at time $t$ has Fourier transform $\exp(t\eta)$ and is thus characterized by $\eta$. The generator (the operator appearing in the Kolmogorov equation) of the (Markov) process $X$ is the pseudo-differential operator with symbol $\eta$. Unfortunately, we will not study this interesting aspect, treated in Chapter 3 of the book. Instead we jump to Chapter 4 where the stochastic integration with respect to Lévy processes is treated and after that jump to Chapter 6 and SDEs. The presentation of the book is quite modern in the sence that it treats stochastic flow properties of the solution processes to SDEs and other more advanced topics, not normally treated in basic books on stochastic analysis. Still, the book is accessible and well written. We will skip almost everything concerning Brownian motion and focus on the jump-part of the process characterized by its corresponding Poisson random measure. The plan is to have weekly meetings and that the participants of the group lecture the material. Active participants will gain 7.5 credit for the course.   Course Homepage