Courses in the Department of Mathematics

MA411: Probability and Measure 2009/10

General information

Course description

Calendar entry for this course

Course Materials on Moodle

General information about MA411: Probability and Measure

Lectures

This is a half unit course, with lectures in the Michaelmas Term. For timetabling information for this course, please see: http://www.lse.ac.uk/admin/timetables/confirmed/module_sessional/ma/27.htm  

There are also revision lectures in the Summer Term.

Classes will start in Week 2 of the Michaelmas Term and will continue into Week 1 of the Lent Term.

Reading

Full lecture notes will be made available. There are a number of books that you may find helpful:

• J.S. Rosenthal, A First Look at Rigorous Probability Theory, World Scientific, 2000
• G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, OUP, 2001
• D. Williams, Probability with Martingales, CUP, 1991
• N.H. Bingham and R. Kiesel, Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives, Springer, 2004

Exercises

Exercises will be set every week in the lectures. All handouts will appear on the Moodle course page at the appropriate time: if you miss one in a lecture, this is the first place to look. Homework should be handed in on Thursdays in the class teacher's homework box (on the ground floor of Columbia House). Please label your work with your name and course number. No late work will be accepted, but partial solutions or solutions to only some of the questions are welcome.

Marked work will be returned and discussed in classes.

Course description of MA411: Probability and Measure

The Calendar entry gives an overview of the course content. If you are intending to take the course, we encourage you to talk to us: we might be amenable to (reasonable!) suggestions about what material should be covered. Let us stress that this is a Pure Mathematics course, and a background in Pure Mathematics is essential, especially familiarity with abstract reasoning and proving theorems in a formal manner. A degree of prior knowledge of analysis would also be of help, in particular familiarity with integrals.

Aims

The course is designed to

• enable you to understand the formal basis for probability theory, and to acquire further skills in proof techniques, reasoning within an abstract system and mathematical modelling; 
• to prepare you for further study in disciplines requiring the use and understanding of probability theory.

Learning outcomes

Having followed MA411, you should have acquired

• knowledge of the formal language of probability theory
• understanding of the precise meaning of concepts such as probability space, probability measure, random variable, expectation and conditional expectation in a very general setting
• understanding of the nature of argument within an abstract system of concepts
• the ability to use probability theory to model simple real-life problems
• the ability to learn in unfamiliar contexts, and to express arguments clearly and accurately.

Course Content

Mathematical theory of probability can be used to model real-world phenomena that are perceived as random, for instance games of chance, the evolution of prices on the stock market, or the performance of telephone networks and the internet. It incorporates and formalises notions implicit in common understanding of uncertainty and risk. This course aims to explain the foundations of rigorous probability theory. The emphasis is on those aspects of the theory commonly used in advanced analytical models in economics and finance.

We cover the following topics:

• Probability spaces and probability measures
• Random variables
• Integration and expectation
• Convergence of random variables
• The Radon-Nikodym Theorem
• Stochastic processes
• Martingales
• Brownian motion
• The Ito integral

Course materials

Lecture notes and exercises will appear on the Moodle course materials page in due course. Comments, criticisms, suggestions for future development, and details of any errors and misprints should be sent to Dr Arne Lokka.