Universität Duisburg-Essen
Fakultät Mathematik, Campus Essen
Anita Winter

2.October 2015

Course (4SWS+2SWS)
Winter Semester 2015/16
Classes: Monday, 10.15-12.00 (S-U 3.03) and Wednesday 10.15-12.00 (S-U 3.03)
& Exercise Session: Wednesday 16.15-17.45 (S-U 3.03)

Probability Theory II


  1. Weak convergence
  2. Portmanteau theorem, Helly's theorem, tightness, Prorohov's theorem, deFinetti's theorem
  3. Characteristic Functions und Laplace-Transforms
  4. Separable and convergence determining classes of functions
  5. Weak limit theorems
  6. Central Limit Theorem, Poisson Convergence
  7. Martingale Theory
  8. Filtrations; Stopping Times; Martingale, Sub- und Supermartingales; Predictability and Doob-Zerlegung; Optional Sampling; Optional Stopping; Martingale Convergence Theorems
  9. Brownian Motion
  10. Stochastic Processes; Gaussian Processes; Construction of Brownian Motion; Strong Markov-Property and Reflection Principle; Increasing points of Brownian motion and random walk paths; Law of iterated logarithm; Skorohod-embedding und invariance principle; Kolmogorov-Smirnov-statistics and the Brownian bridge
  11. Introduction in stochastischen analysis
  12. Ito-integral; Ito-formula; weak and strong solutions of stochastic differential equations; existence and uniqueness theorem

The course is for students with knowledge in measure theoretic probability (Wahrscheinlichkeitstheorie I)

Exercise sessions
The excercise sessions are supported by Wolfgang Löhr.

anita (dot) winter `at' due (dot)de