**Prerequisites**

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".

**Aim of the course**

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.

**Rules about Homework/Exam**

Homework exercises (compulsory and in principle weekly!). The deadline for each assignment is two weeks after the announcement, unless stated otherwise.

The final grade will be based on the homework exercises (40%) and an oral exam (60%).

The minimum required average grade for homework is 5, however the lowest homework grade is not counted in the homework average.

**Lecture Notes / Literature**

On the webpage of the instructor: http://www.math.leidenuniv.nl/~spieksma/onderwijs.html

you can find background literature.

Relevant background definitions and results are summarised in the Background notes, see below.

The lecture notes are an extended version of the Lecture Notes "An introduction to Stochastic Processes in Continuous Time" by Prof. H. van Zanten, see below. Warning: the lecturer still plans to update Chapters 3, 4 and 5, so it may not be sensible to print these chapters!

**Lecturer**: F. Spieksma (UL)