Measure theoretic probability
2005-2006

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 as ps and pdf files.

Examination

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

Student presentations

Students are required to prepare a 20 minutes presentation, see this year's schedule.

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 13. During the first two weeks there will be three hours of classes per week. No class on October 25!!. The schedule, as well as the detailed programme below, may change during the course. Please, always check these items, to see the latest information. Last lecture (on December 13) starts at 11.15.

Reimbursement of travel costs

Students who are registered in a master program in Mathematics at one of the Dutch universities can claim their travel expenses, see the rules.

Programme

1
 Class: Williams sections 1.2 - 1.6, 1.8 (partly), 1.9, 1.10, A1.2 - A1.4
Homework: From Chapter 1: 1, 2, 3, 6 (see ps and pdf files above)
2
 Class: Williams chapters 2, 3 (with the exception of 3.13 and 3.14), 4.1, 4.2, 4.3
Homework: Make four exercises from the set "Chapter 2: 1, Chapter 3: 1, 3, 4, 5, Chapter 4: 2"
3
 Class: Williams sections 5.1-5.7, 5.9, 5.10 (partly), and (very brief) 6.1, 6.2
Homework: Make four exercises from the set "Chapter 4: 3 and Chapter 5: all"; read sections 6.1 and 6.2 to become familiar with the new notation.
4
 Class: Williams sections 6.6, 6.7, 6.12 (briefly), 6.13, 8.1, 8.2 (partly)
Homework: Make three exercises from "Chapter 6: 4, 6 and Chapter 8: 1, 4"; read sections 1 and 2 from the (Radon-Nikodym lecture notes) as a preparation for next week.
5
 Class: Williams sections 8.3, 8.4, 8.5, Radon-Nikodym (lecture notes) sections 4, 5 for positive measures
Homework: Make four exercises from "Chapter 8: 2, 3 and Radon-Nikodym: 7.5, 7.6, 7.9, 7.11"
6
 Class: Williams sections 9.1, 9.2 (via Radon-Nikodym), 9.4, 9.6, 9.7, 9.8, 9.10 (slightly different)
Homework: Read section 9.9; make four exercises from "Chapter 9: "2, 3, 4, 5, 6, 7"
7
 Class: Williams sections 10.1-10.10, 11.1-11.5, 11.7
Homework: Make four exercises from "Chapter 10: 1, 2, 3 and Chapter 11: 1, 2, 4"
8
 Class: Williams chapter 13 and sections 14.1, 14.2, 14.4
Homework: Make four exercises from "Chapter 13: 1, 2, 3, 4, 5, 7"
9
 Class: Williams sections 14.3, 14.5-14.6, 14.10-14.11, 14.12
Homework: Make four exercises from "Chapter 14: 2, 4, 5, 6, 7, 8"
10
 Class: Williams sections 17.1 - 17.5 (last section very briefly)
Homework: Make four exercises from "Chapter 17: 1, 2, 3, 6, 7, 8"
11
 Class: "Central limit theorem" (lecture notes), sections 1,2,3
Homework: From the lecture notes exercise 4.7 and three exercises out of 4.1, 4.2, 4.3, 4.4
12
 Class: Williams, sections 16.1-16.3, 16.6, 18.1, 18.4 and "Central limit theorem" (lecture notes), section 4, but very briefly.
Homework: From the lecture notes exercises 4.5, 4.6 and 4.8 or 4.9
13
 Class: Weak convergence and Brownian Motion (lecture notes in ps and in pdf)
Homework: None


Links

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