Stochastic Analysis

Brownian motion: construction and path properties, reflection principle. Stochastic integrals with respect to Brownian motion and Itô's formula. Stochastic integrals with respect to continuous martingales, cross-variation and Itô's product rule. Stochastic differential equations, weak and strong solutions. Lévy' s Theorem, Girsanov's Theorem and applications. Donsker's invariance principle.

After successful completion of the module, students are able to:
- define Brownian motion and apply basic calculations
involving Brownian motion
- understand fundamental results such as the reflection
principle for Brownian motion, Lévy's Theorem and
Donsker's invariance principle
- understand the basics of stochastic integration
- apply Itô's formula
- understand the basics of stochastic differential equations
- apply change-of-measure techniques.

MA2409 - Probability Theory

lecture, exercise module
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

blackboard, assignments

F. den Hollander, M. Löwe, H. Maassen (1997): Stochastic Analysis, Lecture Notes, University of Nijmegen,
Netherlands.
P. Mörters, Y. Peres (2010): Brownian Motion, Cambridge University Press, New York / Melbourne / Madrid / Cape Town / Singapore / Sao Paulo / Delhi / Dubai / Tokyo

The module examination is based on a written exam (60-90 minutes). Students have to know theoretical foundations of Brownian motion, Lévy's Theorem and Donsker's invariance principle. They are able to understand the basics of stochastic integration and stochastic differential equations and can apply Itô's formula.