Measure theoretic probability
2007-2008

Aim

To provide an introduction in the basic notions and results of measure theory and how these are used in probability theory.

Contents

During the course the measure theoretic foundations of probability theory will be treated. Key words for the course are: measurable space, limit theorems for Lebesgue integrals, product measures, 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.

Prerequisites

Knowledge at the level of for instance Richard T. Durrett, The Essentials of Probability and the first seven chapters of Walter Rudin, Principles of Mathematical Analysis.

Literature

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

Examination

Take home exercises (please, write in english) and oral exam. Homework assignments have to be made in pairs. Deadlines: solutions have to be handed in within one week! The dates for the oral exam will be fixed upon request. Important to know is that January 14-17 and January 21-23 are NOT available and that most likely the same holds for January 25-29. Still, you are strongly advised to take the oral exam before the courses of the spring semester start.

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, Wednesdays 10.15-13.00, room S.201 (Vrije Universiteit, De Boelelaan 1081a), the course will start on September 12. During the first two weeks there will be three hours of classes per week. There will be no lecture on November 21.

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

Please check the programme regularly, weekly updates are expected

1
 Class: Williams sections 1.2 - 1.6, 1.8 (partly), 1.9, 1.10, A1.2 - A1.4 ,2.1, 2.2, and parts of 2.3, 2.6, 2.9
Homework: From Chapter 1: 1, 2, 3, 6 (see ps and pdf files above)
2
 Class: Williams sections 2.3, 2.6 - 2.8, chapter 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: 1"
3
 Class: Williams sections 5.1-5.7, 5.9, 5.10 (hardly)
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 and a sketch of 8.2
Homework: Make three exercises from "Chapter 6: 3, 4, 6 and Chapter 8: 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 (all here and there rather sketchy)
Homework: Make two exercises from "Chapter 8: 1, 3, 5, 6" and two from "Radon-Nikodym: 7.5, 7.9, 7.11" (On October 11 I added an explanatory sentence at the end of Exercise 7.5, without changing the exercise.)
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), section 2 and Williams sections 16.1 and 16.2.
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 16.4a, 16.6, 18.1, 18.4 and "Central limit theorem" (lecture notes), section 3.
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 .