University of Duisburg-Essen

Faculty of mathematics, Campus Essen

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

Lecture with exercises (2SWS+1SWS, 6ECTS)
in the
Winter term 2015/16

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

Exercises: Thursdays 14.00-15.00 (room WSC-S-U-3.03)

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

Exercises: Thursdays 14.00-15.00 (room WSC-S-U-3.03)

**Markov processes and martingale problems**

*Target audience:*

This lecture is an in-depth module stochastics (Vertiefungsmodul Stochastik)
for the master's programme (6ECTS).
The work load is about 150 hours (45 hours of presence time).

*Content:*

Markov processes are memoryless stochastic processes.
These can be characterized in many ways.
Due to many advantages (see below), we focus on the description with
martingale problems.
Given an infinitesimal description of the process (known as generator
which is just a linear mapping on a function space),
a stochastic process solves the martingale problem, if a test function
applied to the process and centered by a bounded variation process is a martingale.
The central advantage of this characterization of a Markov process is
that the beautiful martingale theory can be applied and then
weak convergence of Markov processes can be elegantly derived
from converging martingales.
We will
show how successful martingale problems are for proving
existence of very general multidimensional diffusion processes,
so-called super-processes (measure-valued branching processes)
and
tree-valued Markov processes.
The main goal of the lecture is to equip students with a fundamental
tool box for studying stochastic processes.

*Begin:*

The first lecture will be on Thursday, October 22, 2015.
The exercises start on Thursday, October 29, 2015.

*Requirements:*

Students are required to be familiar with the lectures
'Wahrscheinlichkeitstheorie 1' and 'Wahrscheinlichkeitstheorie 2'
(or its contents, e.g., through Klenke's book 'Probability theory').

*Exercises:*

You will find all exercise sheets on the moodle course page.

*Language:*

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

*Literature:*
Ethier and Kurz 1986: Markov processes

*Moodle course:*

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

*Exam:*

The exam will be on February 11, 2016 from 14:15-16 in room WSC-S-U-3.03