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).

Begin:

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

Content:

  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


Requirements:
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

Exercises:
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.

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

Literature: