Measure theoretic probability
2004-2005

Contents

During the course the measure theoretic foundations of probability theory will be treated. Key words for the course are: random variables, distributions of random variables, different convergence concepts for random variables (convergence in probability, weak convergence, convergence in p-th mean) and relations between them, uniform integrability, conditional expectation and conditional distribution. All these topics will be present in the treatment of martingale theory in discrete time. Finally, the existence of Brownian motion is proved.

Literature

D. Williams, Probability with martingales, Cambridge University Press (BUY IT!) and additional lecture notes.
Exercises available in ps and in pdf

Examination

Homework exercises (please, write in english) and oral exam

Student presentations

Students are required to prepare in pairs a 20 minutes presentation, see the schedule for this year.

People

Lectures by Peter Spreij, homework assistance by Shota Gugushvili.

Schedule

Fall semester, Tuesdays 10.15-13.00, room S205 (Vrije Universiteit!), the course will start on September 7. During the first four weeks there will be three hours of classes per week. There will be no classes on October 5 and October 26. Last class on December 7 begins at 11.15.

Programme

1
 Class: Williams sections 1.2 - 1.6, 1.9, 1.10, A1.2 - A1.4
Homework: From "Week 1": 1,2,3,6 (see ps and pdf files above)
2
 Class: Williams sections 1.7, 1.8 (partly), chapters 2, 3 (with the exception of 3.13), A3.1, A3.2 (partly), sections 4.1, 4.2
Homework: 4 exercises from "Week 1:4" and "Week 2"
3
 Class: Williams sections 5.1-5.7, 5.9, 5.10, 5,12, 5.13, 6.1, 6.2, 6.6, 6.7, 6.13 (partly)
Homework: Exercise 4 from "Week 3" and 3 other exercises from "Week 3"
4
 Class: Williams chapter 8, Radon-Nikodym lecture notes (in ps and in pdf) sections 2, 3 for positive measures
Homework: Exercises 1,2 from "Week 4", exercises 5.7, 5. 9 from the lecture notes.
5
 Class: Williams sections 9.1, 9.2 (via Radon-Nikodym), 9.4, 9.6, 9.7, 9.8, 9.10 (slightly different)
Homework: 4 exercises from "Week 5".
6
 Class: Williams sections 10.1-10.10, 11.1-11.5
Homework: 4 exercises from "Week 6"
7
 Class: Williams chapter 13 and sections 14.1, 14.2, 14.4
Homework: 4 exercises from "Week 7"
8
 Class: Williams sections 14.3, 14.5-14.6, 14.10-14.11
Homework: 4 exercises from "Week 8"
9
 Class: Williams sections 17.1 - 17.4
Homework: 4 exercises from "Week 9"
10
 Class: "Pollard" sections II.3-4 (copies of these sections will be provided)
Homework: Exercises 4 and 5 and two other exercises from "Week 10"
11
 Class: Weak convergence and Brownian Motion (lecture notes in ps and in pdf)
Homework: 4 exercises from "Week 11"
12
 Class: Williams, sections 16.1-16.3, 16.6, 18.1, 18.4
Homework: nothing?


Links

Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics
Dutch Master Program in Mathematics .