University of Duisburg-Essen

Faculty of mathematics, Campus Essen

Prof. Martin Hutzenthaler

February 2, 2016

Lecture with exercises (2SWS+1SWS, 6ECTS)
in the
Summer term 2016

Lecture: Mondays 13.30-15.00 (room WSC-N-U-3.04)

Exercises: Mondays 12.15-13.00 (room WSC-N-U-3.04)

Lecture: Mondays 13.30-15.00 (room WSC-N-U-3.04)

Exercises: Mondays 12.15-13.00 (room WSC-N-U-3.04)

**Stochastic differential equations and their approximations**

*Target audience:*

This lecture is an in-depth module stochastics (Vertiefungsmodul Stochastik)
for the master's programme (6ECTS).
The work load is about 180 hours (45 hours of presence time).

*Begin:*

The first lecture will be on Monday, April 11, 2016.
The exercises start on Monday, April 18, 2016.

*Content:*

- Reminder: Martingale theory definition and properties of (local) martingales and semimartingales
- Ito Integrals Construction; Properties; Ito formula; martingale representation 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
- Numerical approximations of SDEs

*Requirements:*

Students are required to be familiar with martingales and with Brownian motion
e.g., through the lecture 'Probability theory II' or through
chapters 1-10,21 in Klenke's book

*Exercises:*

You will find all exercise sheets on the moodle course 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