Section: Course Materials | B8.1 Probability, Measure and Martingales (2024-25)

Lecturer: Profile: Jan Obloj

Course information

General Prerequisites:

The course is self-contained but subsumes both Part A Probability and Part A Integration. It relies strongly on the intuition and knowledge built up in those two courses and both are strongly recommended.

Course Term: Michaelmas

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: H

Assessment Type: Written Examination

Course Overview:

Probability is both a fundamental way of viewing the world and a core mathematical discipline. In recent years there has been an explosive growth in the importance of probability in scientific research. Applications range from physics to neuroscience, from genetics to communication networks and, of course, finance.
This course develops the mathematical foundations essential for more advanced courses in probability theory. The first part of the course develops a more sophisticated understanding of measure theory and integration, first seen in Part A Integration. The second part focuses on key probabilistic concepts: independence and conditional expectation. We then introduce discrete time martingales and establish results needed to study their behaviour. This prepares the ground for continuous martingales, studied in B8.2, which are the cornerstone of stochastic calculus.

Learning Outcomes:

The students will learn about measure theory, random variables, independence, expectation and conditional expectation, product measures, filtrations and stopping times, discrete-parameter martingales and their properties and applications.

Course Synopsis:

Measurable sets, \(\sigma\)-algebras, \(\pi\)-\(\lambda\) systems lemma. Random variables, generated σ-algebras, monotone class theorem. Measures: properties, uniqueness of extension, Carathéodory's Extension Theorem; measure spaces, pushforward measure, product measure.
Independence of events, random variables and \(\sigma\)-algebras, relation to product measures.
The tail \(\sigma\)-algebra, Kolomogorov's 0-1 Law, lim sup and lim inf of a sequence of events, Fatou and reverse Fatou Lemma for sets, Borel-Cantelli Lemmas.
Integration and expectation, review and extension of elementary properties of the integral and convergence theorems [from Part A Integration for the Lebesgue measure on \(R\)]. Radon-Nikodym Theorem [without proof], Scheffé's Lemma. Integration on product space, Fubini/Tonelli Theorem. Different modes of convergence and their relations. Markov’s and Jensen’s inequalities. \(L^p\) spaces, Holder’s and Minkowski’s inequalities, completeness. Uniform integrability, Vitali’s convergence theorem.
Conditional expectation: definition, properties, uniquness. Conditional convergence theorems and inequalities, link with uniform integrability. Orthogonal projection in \(L^2\), existence of conditional expectation.
Filtrations and stopping times. Examples and properties. \(\sigma\)-algebra associated to a stopping time.
Martingales in discrete time: definition, examples, properties, discrete stochastic integrals. Doob’s decomposition theorem. Stopped martingales and Doob's Optional Sampling Theorem. Maximal and \(L^p\) Inequalities, Doob's Upcrossing Lemma and Martingale Convergence Theorem. Uniformly integrable martingales, convergence in \(L^1\). Backwards martingales and Kolmogorov's Strong Law of Large Numbers.

Section outline

Course materials include: Live lectures videos, Panopto pre-recorded videos, Lecture Notes, Problem Sheets, an R code, additional slides.
Lectures will be streamed live and available to watch later.
In course's Panopto folder, you will find a subfolder MT2020. In there are 26 videos of my 2020 pre-recorded lectures of various length. They cover specific sections of the lecture notes and follow a uniform naming convention: video XY is the Yth video on the contents of Chapter X. All videos are accessed via Panopto and all have attached PDFs with the complete set of notes made during the video. Please note: these videos are from MT2020 and are only an aid, they may miss some examinable material while covering some non-examinable material.
Lecture Notes will be updated as the term goes on - I remove typos when/if those are spotted and I try to add on more of the non-examinable material.
There are four problem sheets, see below, and each is clearly linked to the relevant chapter in the Lecture Notes and the videos.
Finally, I post here some additional materials, e.g., R Code for SSRW examples (covered in 02 - Introduction), notes on percolation or applications further afield. Stay tuned!
PS. At this stage of your education, I think it is useful to include materials which allow those who are interested to undertake guided excursions beyond the syllabus. This is reflected in the length of the lecture notes and wealth of other material here. I should stress that the examinable material is summarised in the syllabus and covered in the lectures.
- Select activity Lecture Notes
  
  Lecture Notes File
- Select activity Problem Sheet 1
  
  Problem Sheet 1 Assignment
  
  The first problem sheet out of four. The questions correspond to the material covered in Chapters 1 & 2.
  
  It contains three sections:
  A) warm-up problems (with solutions) that will not be discussed in class unless asked for
  B) main part with problems for handing in
  C) additional problems worth trying. Some of these may be discussed in classes.
  
  I will update the sheet later on making solutions to problems in part C) visible.
- Select activity Problem Sheet 2
  
  Problem Sheet 2 Assignment
  
  The second problem sheet out of four.
  The problems correspond to the material covered in Chapters 3, 4 & 5.1-5.3.
  
  It contains three sections:
  A) warm-up problems (with solutions) that will not be discussed in class unless asked for
  B) main part with problems for handing in
  C) additional problems worth trying. Some of these may be discussed in classes.
  
  I will update the sheet later on making solutions to problems in part C) visible.
- Select activity Sheet 3
  
  Sheet 3 Assignment
  
  The third problem sheet out of four.
  It may appear longer than usual but many problems are simple and/or short.
  The problems correspond to the material covered in Chapters 6, 7 & 8.1. The coverage of 8.1 is only via checking the definitions (two problems in Part A). For some of you this may come before the material is covered in the lectures - you can either look up the definitions in the notes ahead of lectures or do these problems afterwards.
  
  It contains three sections:
  A) warm-up problems (with solutions) that will not be discussed in class unless asked for
  B) main part with problems for handing in
  C) additional problems worth trying. Some of these may be discussed in classes.
  
  I will update the sheet later on making solutions to problems in part C) visible.
- Select activity Sheet 4
  
  Sheet 4 Assignment
  
  The fourth problem sheet out of four.
  The problems correspond to the material covered in Chapters 8 & 9.
  
  It contains three sections:
  A) warm-up problems (with solutions) that will not be discussed in class unless asked for
  B) main part with problems for handing in
  C) additional problems worth trying. Some of these may be discussed in classes.
  
  I will update the sheet later on making solutions to problems in part C) visible.
- Select activity R Code for SSRW examples
  
  R Code for SSRW examples File

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Section outline