The course deepens the understanding of concepts, methods and tools from stochastic analysis; specifically continuous time Markov chains and Levy processes.
It introduces two main mathematical concepts, namely continuous time Markov chains and Levy processes. Markov chains play a prominent role in financial modelling where they are often used for regime-switching phenomena. Levy processes are a natural generalisation of Brownian motion and many concepts for Brownian motion can be generalised to Levy processes. More importantly, they give a key understanding of semimartingales in general which somewhat locally behave like Levy processes.
This course starts with an outline of its goals. Then, continuous time Markov chains are discussed. We develop transition graphs, transition laws, semigroups and generators for them. Subsequently, we start to deal with Levy processes, discuss its Levy-Ito decomposition and introduce the so-called Levy triplet. The latter is a minimal probabilistic description of the Levy process at hand.