Universität Duisburg-Essen

Fakultät Mathematik, Campus Essen

Prof. Martin Hutzenthaler

Course (4SWS+2SWS, 9ECTS)

Winter Semester 2016/17

Lectures: Mondays, 12.15-14.00 (WSC-S-U-3.02) and Wednesdays 10.15-12.00 (WSC-S-U-3.03)

Exercise classes: Wednesdays 12.15-14.00 (WSC-S-U-3.03)

**Probability Theory II**

Winter Semester 2016/17

Lectures: Mondays, 12.15-14.00 (WSC-S-U-3.02) and Wednesdays 10.15-12.00 (WSC-S-U-3.03)

Exercise classes: Wednesdays 12.15-14.00 (WSC-S-U-3.03)

*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).

*Requirements:*

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

*Contents:*

- Weak convergence Portmanteau theorem, Helly's theorem, tightness, Prorohov's thoerem, deFinetti's theorem
- Characteristic functions and Laplace-transforms Separable and convergence determining classes of functions
- Weak limit theorems Central Limit Theorem, Poisson Convergence
- Martingale Theory Filtrations, Stopping times, martingales, predictability and Doob-decomposition, optional sampling and optional stopping theorem, martingale convergence theorem
- Brownian motion
- 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
- Introduction to stochastic analysis Ito-intgral, ito-formula, weak and strong solutions of stochstic differential equations, existence and uniqueness theorem

*Exercises*

The exercise sessions are supported by
Thomas Kruse.

*Moodle-course*

You will find the lecture notes and exercise sheets on
tba

*Literature:*

- Achim Klenke; Probability theory, Springer.
- Ulrich Krengel; Einfuuml;hrung in die Wahrscheinlichkeitstheorie und Statistik, Springer.

*Begin:*

- The lectures start on October 17, 2016. The exercises start on October 19, 2016.