This course is an introduction to the theory of continuous-time stochastic processes. We intend to treat some classical, fundamental results and to give an overview of two important classes of processes.
These processes are so-called martingales and Markov processes. The main part of the course is devoted to developing fundamental results in martingale theory and Markov processes theory, with an emphasis on the interplay between the two worlds.
The particular applications we will study are Markov jump processes,and Brownian motion.
Markov jump processes already exhibit the structural properties
of the general Markov processes that we will discuss.
Brownian motio is an important process that is both a martingale and a Markov process. We will derive a number of its
fascinating properties.
If there is any time left, we can study other special classes of
Markov processes. For instance Brownian motion in higher dimensions,
diffusions, L\‘e vy processes, continuous time Markov chains, counting
processes.
Prerequisite
We assume the students have a solid background in the measure theoretic foundations of probability theory. Prior knowledge of conditional expectation and discrete-time martingales is not strictly necessary, but strongly recommended.
Literature
Lecture Notes by Harry van Zanten
Methods of instruction
3 hours lecture per week
Examination
<td valign="\"top\""></td><td valign="\"top\"">oral examination and homework</td>### Remarks
this course is part of the Master Programme in Mathematics.
Links
<font color="#0000ff">Webpage of the course</font>