Stochastic Processes

This course is an introduction to the theory of continuous-time stochastic processes. We plan to treat a number of classical results and to introduce 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 (first in discrete time and then in continuous time) as well Markov process theory, with an emphasis on the interplay between the two. Special features of Markov processes that we aim to discuss, are
the strong Markov property and (as an extension to previous editions of this course) 
explosion phenomena as well as limit behaviour.

As a main illustration of the theory, we will study the fascinating properties of Brownian motion, an important process that is both a martingale and a Markov process.
We also plan to discuss applications to processes on a countable state space, such as the Poisson process, and birth-death processes, which play an important role in queueing theory.

If there is any time left, we can study other special cases of Markov processes. For instance
Brownian motion in higher dimensions, diffusions and Levy processes, countbale state space Markov processes and counting processes.

Homework exercises (compulsary and in principle weekly!) and oral exam. The final grade is based on the results of both homework and oral exam.

The deadline for each assignment is two weeks after the announcement, unless stated otherwise.