1. Prerequisites

We assume prior knowledge of elementary measure theory, in a probabilistic context. It is recommended to take the course `Measure Theoretic Probability' before the SP course. A good reference is Williams' book `Probability with Martingales' or you can download Peter Spreij's lecture notes (https://staff.fnwi.uva.nl/p.j.c.spreij/onderwijs/master/mtp.pdf)

2. Course Description

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 process 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 some applications in queueing theory.

The course on Stochastic Integration is a recommendable companion course.

3. Examination

This course will contain homework, a presentation moment and a final oral exam. These contribute to the final mark as with the following weights

homework 40%, presentation 10% and final exam 50%

4. Literature

The course is based on lectures notes written by Harry van Zanten in 2005. The lecture notes are constantly being revised. The lectures still want to browse throught them before the course starts, so we recommend not to print more than the first chapter for the time being.

Also we will use a pdf with background material. See below. LN=lecture notes, BN= background notes.

For further reading you can consult the following books, the level of which is far more technical than the lecture notes:

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. Rogers and D. Willimans, 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, Markov Processes: Characterization and Convergence, 1986, J. Wiley and Sons.

R.M. Dudley, Real Analysis and Probability, 2002, Cambridge University Press.

5. Lecturers

Dr F. Spieksma (spieksma at math.leidenuniv.nl)

6. Recordings

The lectures will be recorded. you can find these at

https://vimeo.com/showcase/10950409

using the password 8012

- Docent: Azadeh Parvaneh Ziabari
- Docent: Floske Spieksma