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. Special features of Markov processes that we aim to discuss, are the strong Markov property, the generator, explosion phenomena as well as limit behavior. The latter provided time permits.
As a main illustration of the theory, we shall 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 queuing theory.
If there is any time left, we can study other examples of Markov processes. For instance, Brownian motion in higher dimensions, diffusions and Levy processes, countable state space Markov processes and counting processes.
We assume that students have a solid background in the measure theoretic foundations of probability theory, as well as some knowledge of discrete-time martingales.
A crucial preparatory course is the Mastermath course "Measure Theoretic Probability".