University of Duisburg-Essen
Faculty of mathematics, Campus Essen
Martin Hutzenthaler (martin (dot) hutzenthaler 'at' due (dot)de)
Anton Klimovsky (ak 'at' aklimovsky (dot) net)

8.September 2014

Lecture with exercises (4SWS+2SWS) in the Winter term 2014/15
Lecture: Tuesdays, 10.15-12.00 (room WSC-S-U-3.03) and Thursdays 10.15-12.00 (room WSC-S-U-3.03)
Exercises: Tuesdays 8.30-10.00 (room WSC-S-U-3.03)

Probability theory II

Target audience:

This lecture is both for Masters students and Bachelor students. This lecture is an 'Aufbaumodul Stochastik' (9ECTS). The work load is about 270 hours (90 hours of presence).


The first lecture will be on Tuesday, October 14, 2014. The exercises start on Tuesday, October 14, 2014, with a short repetition.


  1. Martingale theory
  2. Optional sampling; optional stopping; Doob's inequality; martingale convergence theorem; backward martingales; de Finetti's theorem
  3. Convergence of measures
  4. Weak and vague convergence; Prohorov's theorem
  5. Product measures
  6. Theorem of Ionescu-Tulcea; consistent families; Kolmogorov's extension theorem
  7. Central limit theorem
  8. Theorem of Stone-Weierstrass; characteristic functions; the central limit theorem
  9. Brownian motion
  10. Stochastic processes; Gauss processes; construction and path properties of Brownian motion; strong Markov property and reflection principle; law of the iterated logarithm; Skorohod's embedding theorem
  11. Poisson point processes

Students are required to be familiar with measure theory and conditional expectations, e.g., through the lecture 'Wahrscheinlichkeitstheorie 1' or through chapters 1-8 in Klenke's book

Anton Klimovsky supervises the exercises. You will find all exercise sheets on this web page.

Moodle course:
There is a moodle course on which everyone should register. News and messages will be sent over this moodle platform.

The language both in the lecture and in the exercises will be English or - if preferred - German.