# Martingale Theory with Applications 3

## Unit aims

To stimulate through theory and examples, an interest and appreciation of the power of this elegant method in probability theory. And to lay foundations for further studies in probability theory.

## Unit description

The theory of martingales is of fundamental importance to probability theory, statistics, and mathematical finance. This unit is a concise introduction of the basic concepts, results and examples of this powerful and elegant theory.

## Relation to other units

Probability 2 has introduced Martingales, but only covers the most basic of results, mostly without rigorous proofs. This unit will prove most of the results in a rigorous measure-theoretic fashion, and will be essential for students who wish to go on to study post-graduate level probability theory. In particular, students will find the understanding of material in this unit very helpful in other related units, such as Financial Mathematics (MATH35400) and Further Topics in Probability 3 (MATH30006).

## Learning objectives

To gain an understanding of martingales, and to be able to formulate problems in probability/statistics theory in terms of martingales. Students will also gain more experience in writing proofs, thus laying the foundation for future studies in probability theory at a post-graduate level.

## Transferable skills

Formulation of probability/statistics problems in terms of martingales. Better ability in writing proofs.

## Syllabus

Introduction to measure spaces; conditional expectation; definition of martingales; optimal stopping theorem; martingale convergence theorem; L^2 martingales; Doob decomposition; uniformly integrable martingales; applications of martingales to areas such as finance.

## Reading and References

Williams, D., Probability with Martingales (CUP)

A.N. Shyriaev: Probability (Second Edition, Springer)

**Unit code:** MATH36204

**Level of study:** H/6

**Credit points:** 10

**Teaching block (weeks):** 1 (1-6)

**Lecturer:** Dr Marton Balazs

## Pre-requisites

MATH20008 Probability 2

## Co-requisites

None

## Methods of teaching

Lectures and homework assignments. Bi-weekly assignments to be done by the student and handed in for marking.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 80% from a 1 hour 30 minute exam in January
- 20% homework

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.