**Stochastic processes - Fundamentals**

Spring 2016

**For the 2017 course, see the elo mastermath site. Also the latest versions of the lecture notes and background notes will be uploaded there.
**

**Contents**

This course is a measure-theoretic 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. The UVA course on Stochastic Integration taught by Dr Peter Spreij is a recommendable companion course.

**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 2005.

The Lecture Notes are constantly being revised.

The latest version can be found here. Note that we will are continuing to revise it, but corrections, and other additions will be in blue.
We recommend you not to print it before tuesday febr 9, 2016, late.

Background material can be found here. We will frequently refer to it, and we might update it till tuesday febr. 9, 2016, late.

Further reading (note that most books have a higher technical level!):

- R.F. Bass, Stochastic Processes, 2011, Cambridge University Press.
- 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.
- R.M. Dudley, 2002, Real Analysis and Probability, Cambridge University Press.

**People**

The course will be taught by Floske Spieksma (spieksma@math.leidenuniv.nl), the assistents are Khadija Lachhab (email addresses khadijaa_ at live.nl.

Mail address: Maths Institute, Leiden University, PO Box 9512, 2300RA Leiden.

** In principle FS is at the VU on wednesdays (room 540) during the spring semester!**

**Examination**

Homework (weekly in principle!) and an oral exam. Details concerning the oral exam will be provided later.

An overview of your homework grades can be found here.

**Please hand in your homework electronically at spieksma@math.leidenuniv.nl, if possible
Homework system: sometimes you will get the option to choose exercises with a maximum of 10 points. You can choose also to make homework for less points, say a total of 8 points,
then you can maximally get 8 for that homework!**

**Schedule**

Spring semester, Wednesdays **10.15 -13 am** in room ** M639** (**Science building, Vrije Universiteit**).

The course will start on February 10, 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.