Universität Duisburg-Essen
Probability Group at Campus Essen
Wolfgang Löhr and Anita Winter

26th March 2013

Course (4SWS) with integrated problem sessions/Seminar presentations (2SWS)
Summer Semester 2013
Tuesday and Thursday, 10.15-12.00 (Room: WSC-S-U-3.03)

Markov processes: Theory and Examples


Content: The development of any stochastic model involves the identification of properties and parameters that, one hopes, uniquely characterize the stochastic process. In fact, the model may be arrived by some sort of limiting or approximation argument. For example, Brownian motion arises as the suitably re-scaled limit of random walks (with small but frequent jumps). To be in a position to state such a convergence we develop concepts such as the weak topology in the space of probability measures and the Skorokhod topology in the space of cadlag paths. In this class we want to focus on the theory of Markov processes which are an important class of stochastic processes. We also introduce operator semigroups, martingale problems, stochastic equations which provide approaches to the characterization of Markov processes. The theory will be illustrated with many examples. The course is accompanied by a block seminar in which students can present a paper related to the theory.

  1. Topological and probabilistic preliminaries
  2. Cadlag paths and the Skorokhod metric
  3. Markov processes
  4. Weak convergence and the Prohorov metric
  5. Feller semigroup and generators
  6. Harmonic functions and martingales
  7. Convergence of stochastic processes
  8. Strong Markov property
  9. Martingale problems and Dirichlet forms

Präsentationen

Preknowledge: The course is offered in the Master program in Mathematics. It is addressed to students with strong knowledge in measure-theoretic probability theory (Wahrscheinlichkeitstheorie I), preferable Wahrscheinlichkeitstheorie II and/or Funktionalanalysis. Further knowledge about stochastic processes are of advantage. We will adapt the class to the background of the participants. To attract non-German speaking students we are planning to teach this course in English. If all students are comfortable with German, we will teach in German. In any case we mainly follow a script which is written in English. If your are interested in the course, please send an e-mail (anita.winter@uni-due.de) and indicate your pre-knowledge and whether or not it is ok for you to follow the class in English.

Literature: