Stochastic Analysis

Continuous time stochastic processes as a general framework. Brownian motion: construction and path properties. Filtrations, (local) Martingales and stopping times, optional sampling, maximal inequalities and reflection principle. Poisson processes and stable processes. Quadratic and cross variations, Levy’s representation and characterization Theorems, Semi-martingales. Stochastic integrals with respect to Brownian motion and general semi-martingales. Integration rules with emphasis on Itô's formula.
Stochastic differential equations, weak and strong solutions. Girsanov's Theorem and applications. Donsker's invariance principle.

After successful completion of the module, students are able to:
- define Brownian motion, prove basic facts about Brownian motion and apply basic calculations involving Brownian motion.
- understand fundamental results such as Lévy's Theorems, Donsker's invariance principle etc, and apply them in calculations as well as in proofs.
- understand the basics of stochastic integration with respect to various processes. Apply Itô's formula and other tools.
- understand the basics of stochastic differential equations, can verify the (in)existence of strong or weak solutions in standard examples.
- understand and apply change-of-measure techniques.

MA0003 - Analysis 3
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 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

Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, New York: Springer.
P. Mörters, Y. Peres (2010): Brownian Motion, Cambridge University Press, New York / Melbourne / Madrid / Cape Town / Singapore / Sao Paulo / Delhi / Dubai / Tokyo
Le Gall, J. F. (2016). Brownian motion, martingales, and stochastic calculus. Springer.
Revuz, D., & Yor, M. (2013). Continuous martingales and Brownian motion (Vol. 293). Springer Science & Business Media.

The module examination is based on a written exam (60-90 minutes).
Students have to know and understand the theoretical foundations of Brownian motion, the theory of continuous martingales and Lévy's representation and characterisation Theorems. They also need to be able to understand the basics of stochastic integration and stochastic differential equations and to apply Itô's formula and change-of-measure techniques.