Markov processes are stochastic processes without memory. The evolution in the future only depends on the current state of the process but not on the past. Lévy processes are Markov processes with some additional features: They have independent and stationary increments and are stochastically continuous. It turns out that this simple definition already describes a wide range of processes. Their paths may have jumps but are right-continuous and have left limits. Due to this property and to the fact that they have a very nice structure, they are used in all kinds of applications, ranging from the natural sciences to finance -- whenever the model requires to allow for "shocks" at random times. In addition to the wide range of applications, Lévy processes are also a tool for the analysis and numerical solution of non-local partial differential equations.
In this lecture we will give an introduction to Markov processes with a focus on Lévy processes. Topics will include general constructions of stochastic processes, the (strong) Markov property, Poisson processes, compound Poisson processes, decomposition formulas for Lévy processes (the Lévy-Khnitchine and the Lévy-Ito formula), and operator semigroups (from a probabilistic perspective).
- Lehrende(r): Petru Cioica-Licht