Stochastic Processes Spring 2024-2025

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 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.

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)

Teachers 

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

Assistents

• • Daan Zunnenberg (d.w.zunnenberg@math.leidenuniv.nl)

  • Nikolai Kriukov (n.kriukov@uva.nl)

Literature

The course is based on lectures notes written by Harry van Zanten in 2005, and might be regularly  updated. Also we will use a pdf with background material. Both pdfs are included  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.

Examination

Homework (you MAY hand it in in pairs),  and an oral exam  based on a take home (re-) exam (hw grade=40%, exam grade 60%). The minimum  grade for the (re-) exam should be 5; the average homework grade will computed as the average of all homework assignments. Not handing in a homework assignment=0 pts. The weighted average of homework and (re-) exam should be 5.5  in order to pass the course.

An overview of the grades will probably be provided on the elo mastermath site.

Homework should be handed in on the dates specified in the table below.  You can either hand it in during the lecture, or send it via email to the lecturer.  Both written texts as well as a LaTeX pdf are allowed. Written homework should be readable, and, when handed in electronically, in 1 pdf file. Illegible homework=0 pts.

We now plan almost weekly homework sets of 1-2 exercises. This allows you to get more timely feedback.  

Lectures

The lecture will in principle consist of two hours. During the third hour you can work on selected exercises or homework.

Note:

• lecture times are  14-16.45 on wednesdays 

• our usual room is rm NU4B43

• the schedule below is preliminary!

The schedule is below.