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 as Markov process theory, with an emphasis on the interplay between the two.
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 discusses applications such as 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, Levy processes, continuous time Markov chains, counting processes.