University of Duisburg-Essen

Faculty of mathematics, Campus Essen

Martin Hutzenthaler (martin (dot) hutzenthaler 'at' due (dot) de)

Wolfgang Löhr (wolfgang (dot) loehr 'at' due (dot) de)

8.September 2014

Lecture with exercises (2SWS+1SWS)
in the
Winter term 2014/15

Lecture: Thursdays 12.15-14.00 (room WSC-S-U-3.03)

Exercises: Tuesdays 9.15-10.00 (room WSC-S-U-3.03)

Lecture: Thursdays 12.15-14.00 (room WSC-S-U-3.03)

Exercises: Tuesdays 9.15-10.00 (room WSC-S-U-3.03)

**Stochastic differential equations**

*Target audience:*

This lecture is an in-depth module stochastics (Vertiefungsmodul Stochastik)
for the master's programme (5ECTS).
The work load is about 150 hours (45 hours of presence time).

*Begin:*

The first lecture will be on Thursday, October 16, 2014.
The exercises start on Tuesday, October 21, 2014.

*Content:*

- Martingale theory definition and properties of (local) martingales and semimartingales
- Ito Integrals Construction; Properties; Ito formula; martingale representation theorem;
- Ito formula Weak and vague convergence; Prohorov's theorem
- Stochastic differential equations examples of explicitly solvable SDEs; existence and uniqueness; strong and weak solutions; the Yamada-Watanabe pathwise uniqueness theorem; Martingale problems; Markov property
- Applications of SDEs

*Requirements:*

Students are required to be familiar with measure theory and conditional expectations,
e.g., through the lecture 'Wahrscheinlichkeitstheorie 1' or through
chapters 1-8 in Klenke's book

*Exercises:*

Wolfgang Löhr supervises the exercises.
You will find all exercise sheets on this
web page.

*Moodle course:*

There is a
moodle course on which everyone should
register. News and messages will be sent over this moodle platform.

*Language:*

The language both in the lecture and in the exercises will be English
or - if preferred - German.

*Literature:*

- Rogers and Williams; Diffusions, Markov Processes and Martingales 2: Ito Calculus; Cambridge University Press; 2000
- Oksendal; Stochastic differential equations (6th edition); Springer; 2003
- Achim Klenke; Wahrscheinlichkeitstheorie; Springer; 2006
- Achim Klenke; Probability theory; Springer; 2008; translation of the book from 2006