Stochastische Analysis
Stochastic Analysis
Modul MA4405
Diese Modulbeschreibung enthält neben den eigentlichen Beschreibungen der Inhalte, Lernergebnisse, Lehr- und Lernmethoden und Prüfungsformen auch Verweise auf die aktuellen Lehrveranstaltungen und Termine für die Modulprüfung in den jeweiligen Abschnitten.
Modulversion vom SS 2014 (aktuell)
Von dieser Modulbeschreibung gibt es historische Versionen. Eine Modulbeschreibung ist immer so lange gültig, bis sie von einer neuen abgelöst wird.
| verfügbare Modulversionen | |
|---|---|
| SS 2014 | WS 2013/4 |
Basisdaten
MA4405 ist ein Semestermodul in Englisch auf Master-Niveau das im Wintersemester angeboten wird.
Das Modul ist Bestandteil der folgenden Kataloge in den Studienangeboten der Physik.
- Spezialisierung im Elitemasterstudiengang Theoretische und Mathematische Physik (TMP)
| Gesamtaufwand | Präsenzveranstaltungen | Umfang (ECTS) |
|---|---|---|
| 180 h | 60 h | 6 CP |
Inhalte, Lernergebnisse und Voraussetzungen
Inhalt
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.
Lernergebnisse
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.
- 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.
Voraussetzungen
MA2409 - Probability Theory
Lehrveranstaltungen, Lern- und Lehrmethoden und Literaturhinweise
Lehrveranstaltungen und Termine
| Art | SWS | Titel | Dozent(en) | Termine |
|---|---|---|---|---|
| VO | 3 | Stochastic Analysis | Gantert, N. |
Do, 09:15–11:45, 8101.02.201 |
| UE | 1 | Stochastic Analysis (Exercise Session) | Criens, D. Gantert, N. | Termine in Gruppen |
Lern- und Lehrmethoden
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.
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.
Medienformen
blackboard, assignments
Literatur
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
Netherlands.
P. Mörters, Y. Peres (2010): Brownian Motion, Cambridge University Press, New York / Melbourne / Madrid / Cape Town / Singapore / Sao Paulo / Delhi / Dubai / Tokyo
Modulprüfung
Beschreibung der Prüfungs- und Studienleistungen
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.
Wiederholbarkeit
Eine Wiederholungsmöglichkeit wird am Semesterende angeboten.