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# Stochastic Analysis

## Module MA4405

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of WS 2013/4

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2021/2SS 2021SS 2020SS 2014WS 2013/4

### Basic Information

MA4405 is a semester module in English language at Master’s level which is offered in winter semester.

This module description is valid from SS 2014 to WS 2021/2.

180 h 60 h 6 CP

### Content, Learning Outcome and Preconditions

#### Content

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.

#### Learning Outcome

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.

#### Preconditions

MA2409 - Probability Theory

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 3 Stochastic Analysis Gantert, N. Thu, 09:00–11:45, BC2 BC2 3.5.06
UE 1 Stochastic Analysis (Exercise Session) Fernandez, L. Gantert, N. singular or moved dates
and dates in groups
eLearning

#### Learning and Teaching Methods

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.

#### Media

blackboard, assignments

#### Literature

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

### Module Exam

#### Description of exams and course work

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.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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