SP (preliminary version)

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 UvA course on Stochastic Integration taught by Prof dr Peter Spreij 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 

• • Dr L. Avena (l.avena at math.leidenuniv.nl)

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

Assistents (stochproc2018 at gmail.com)

• • Dylan Gonzalez Arroyo

  • Margriet Oomen

  • Benthen Zeegers

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.

Examination

Homework (in principal weekly, you MAY hand it in in pairs) and a written exam (hw grade=40%, exam grade=60%). The minimum  grade for both homework and exam should be 5.5; the average homework grade will computed as the average of all homework assignments, except the one for which you had the lowest grade. Not handing in a homework assignment=0 pts.

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. If you hand it in during the lecture, please also send a copy to the assistent email address. 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.

Exam details  

During the exam you may consult 

1. LN (recent version contains the three materials pdfs, you may also take the old version plus these three pdfs)  and BN

2. personal notes and your homework

3. posted  solutions of some of the HW10 and 11 exercises (will be done June 5, in the morning)

Resit:

1. grade for the course is max(0,6xgrade exam or resit+0,4xgrade hw, 0,5 grade exam or resit+ 0,5 grade hw). Exam grades are on elo under exam rules. Minimum required grades are 5 for both hw and exam (for exam this is according to mastermath rules)

2. I will also put the final hw grades on elo under exam rules, so that you can check your average hw grade. The elo excel file does not contain the correct averages due to the fact that the table is impossible to configure.

3. If you want to do an oral in order to upgrade your grade, please contact Ms Spieksma first

4. Oral exam:

    -- it takes 30-35 minutes and will take place at Mathematisch Instituut, Rm 228, Snelliusgebouw, Niels Bohrweg 1 Leiden

    -- you have to be present 45 minutes before you exam in rm 228. You will get some exercises that you can study 

       during these 45 minutes. These will be part of  the oral. You may consult your own notes during the preparation part.

    -- you should study the exam questions yourself. These will be posted in the next days. No solutions will be provided.

    -- you can enter your name in the google spreadsheets in one of the proposed time slots -- on June 27, 28, and July 2

        and July 3.        

Schedule

Note:

• lecture times are 10.15-13.00

• our usual room is WN M639 on the VU, except  for March 28 (week 13). Then, the room is WN C121.

• the schedule below is not necessarily definite...