Stochastic processes

Spring 2012

Contents
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 general results will then be used to study fascinating properties of Brownian motion, an important process that is both a martingale and a Markov process.
We also plan to study applications like birth-death processes, which is a basic model in queueing theory.
If there is any time left, we can study other special classes of Markov processes. For instance, Brownian motion is higher dimensions,
diffusions, Lévy processes,  counting processes.

Prerequisites
We assume prior knowledge of elementary measure theory, in a probabilitistic context. It is recommended to take the course Measure Theoretic Probability before the Stochastic Processes course. A good reference for self study is Williams' book `Probability with Martingales' or you can download Peter Spreij's lecture notes

Literature
The course is based on lecture notes on stochastic processes written by Harry van Zanten in 2007.

The Lecture Notes are constantly being revised. They have been splitted into three parts, which are the subsequent Chapters to be discussed.
As an appetiser, you can look at the 2011 Lecture Notes. Updates of the subsequent chapters will appear one by one!
The updated versions that we will use are:

• Chapter 1, version January 23 2012;
• Chapter 2, version .... 2012, not available yet;
• Chapter 3, version ... 2012, not available yet;
• Background material version January 23, 2012.

• P. Billingsley, Probability and Measure, 3d Edition, J. Wiley and Sons.
• P. Billingsley, Convergence of Probability Measures, 1999, J. Wiley and Sons.
• L.C.G. Williams and D. Williams, Diffusions, Markov Processes and Martingales, part I, 2000, Cambridge University Press.
• I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 1999, Springer-Verlag.
• S.N. Ethier and Th. G. Kurtz, 1986, Markov Processes Characterization and Convergence, J. Wiley and Sons.

People
The course will be taught by Floske Spieksma (spieksma@math.leidenuniv.nl), assistent Herman Blok.
Mail address: Maths Institute, Leiden University, PO Box 9512, 2300RA Leiden.
In principle FS is at the VU on wednesdays (room 540)!

Examination
Homework (weekly in principle!) and oral exam.
Oral exam: during the exam you will be asked about the theory only, you do not have to know all proofs by heart. Please choose the three theorems that you find most important and prepare a sketch of their proofs. Sections that you do not need to prepare at all will be specified lateron.
An overview of your homework grades can be found in (to be added yet).

Schedule
Spring semester, Wednesdays 10 (!)-13 am in room S655 (Science building, Vrije Universiteit). In week 14, the lecture will take place in room M143!
The course will start on February 8, last lecture will be on May 25.
Please always check this page for the most recent info.
The date between brackets following HWn is the date that the corresponding homework should be handed in. LN stands for `Lecture Notes', and BN for Background Notes'.

Evaluation
At the end of the course you will be asked to evaluate it by filling in a questionnaire.

Programme
This is a preliminary programme that will be updated weekly!