Universitšt Duisburg-Essen
Fakultšt Mathematik, Campus Essen
Prof. Martin Hutzenthaler
Dr. Petru Cioica-Licht

Course (4SWS+2SWS, 9ECTS)
Winter Semester 2019/20
Lectures: Mondays, 12.15-14.00 and Thursdays 12.15-14.00
Exercise classes: Tuesdays 14.15-16.00

Probability Theory II

Target audience:
This lecture is both for Bachelor and Master students (Aufbaumodul Stochastik). The work load is about 270 hours (including 90 hours of presence time).

Students are assumed to be familiar with the contents of the lecture 'Probability Theory I'.


  1. Weak convergence
  2. Portmanteau theorem, Helly's theorem, tightness, Prorohov's thoerem, deFinetti's theorem
  3. Characteristic functions and 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, martingales, predictability and Doob-decomposition, optional sampling and optional stopping theorem, martingale convergence theorem
  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 and invariance principle, kolmogorov-smirnov-statistics and the Brownian bridge
  11. Introduction to stochastic analysis
  12. Ito-intgral, ito-formula, weak and strong solutions of stochstic differential equations, existence and uniqueness theorem

You will find the lecture notes and exercise sheets on Moodle page