MSc Pure Mathematics and Mathematical Logic

Imagine the stochastic process you get if you play a certain fair game (i.e. with zero expected gain) repeatedly and keep track of your total gain over time. This is maybe the most prominent example of a martingale. Martingales make up a very prominent class of stochastic processes, pivotal to the field of (modern) mathematical finance but essential in many other branches of maths where stochastic processes are used as well.

Although in this course you will also do plenty of computations and see examples of applications, the main focus is on developing the theory which includes proving results when we reasonably can. The course starts with the measure theoretic setup of probability spaces on which we will be working, then we go on to study martingales and their fascinating properties, and we conclude by also venturing a tiny bit into the wild world of Lévy processes.  

The unit aims to:

Rigorously introduce and develop the theory of a class of stochastic processes called martingales, as well as study a variety of prominent examples. 
 

• Describe the objects that make up a probability space and apply the classic convergence results
• Evaluate conditional expectations with respect to a sigma algebra and integrals with respect to a measure
• Define a martingale and analyse whether or not a given stochastic process is a martingale
• Apply a suite of classic results for martingales and discuss key insights in their proofs
• Describe and evaluate a number of prominent examples of martingales  
• Define a Lévy process and discuss some prominent examples together with their properties 

1. Measure theoretic basis: probability spaces; integration with respect to a measure; modes of convergence; conditional expectation (with respect to a sigma-algebra); convergence results (monotone convergence, dominated convergence, Fatou’s lemma)

2. Discrete time: definition, examples and basic properties of (super/sub)martingales; seminal results including the martingale convergence theorem, the optional stopping theorem, maximal inequalities and the Doob-Meyer decomposition

3. Continuous time: definition of (super/sub)martingales as well as local martingales; extending the seminal results from 2. to continuous time; studying a number of prominent examples (including Brownian motion, Poisson processes, compound Poisson processes, jump-diffusions), also in the context of some exploration of the class of Lévy processes

Note: MATH37021 Foundations of Modern Probability is recommended because it discusses elements of part 1. of the syllabus in more detail than this course does. However this course does not assume that you have seen that material before. 

The weekly learning cycle for this course consists of:  

1. an asynchronous part where students self-study the materials for the week supported by detailed typeset notes and videos in which the materials are discussed in a ‘talk & chalk’ format;

2. a 2 hour block of contact time devoted to reviewing/discussing key points of the materials (student led) with a focus on addressing student questions, as well as working through a number of examples/exercises (individual/small group initially followed by general discussion)

3. the remaining 1 hour of contact time is a non-compulsory drop-in question&answer session giving students the opportunity to seek individual support and feedback, and/or to work on the remaining exercises for the week with support at hand.

Note: in addition to the weekly feedback opportunities during the classes, prior to the exam students will be able to get individual feedback on their solutions of past exam paper questions (and otherwise) 
 

Take home coursework - 20% 
4 hours (in a 1-week window) 
As soon as marking completed (within Dept deadlines), feedback both individually (on script) as well as class level (common mistakes etc.)

End of semester exam - 80%
3 hours 
Generic feedback available after the exam period 

Detailed lecture notes will be provided 

Probability with martingales    Williams, David, 1938 Apr. 9-    Cambridge University Press    1991    ISBN: 052140455

Probability and random processes    Grimmett, Geoffrey.    Oxford University Press    2001    ISBN: 0198572239

Essentials of Integration Theory for Analysis    Stroock, Daniel W.    Springer New York    2011    ISBN: 9781461411352

A concise introduction to the theory of integration    Stroock, Daniel W.    Birkhäuser    1994    ISBN: 0817637591

Scheduled activities include 22 hours review+tutorial (in blocks of 2 hours per week) plus 11 hours drop-in (1 hour per week)