# Unit information: Martingale Theory with Applications 3 in 2021/22

Unit name Martingale Theory with Applications 3 MATH30027 20 H/6 Teaching Block 1 (weeks 1 - 12) Dr. Balazs Not open MATH20008 Probability 2 None School of Mathematics Faculty of Science

## Description including Unit Aims

Unit aims

To lay foundations for further studies in probability theory. To stimulate through theory and examples, an interest and appreciation of the power of the elegant method of martingales in probability theory.
Unit description

First, rigorous foundations and essential tools of probability theory are introduced; various modes of convergence of random variables (almost sure, L^p, in probability, weak) and the connections between them are presented. We then turn to the theory of martingales which 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 and Further Topics in Probability (for which this unit is a prerequisite).

Intended learning outcomes

To gain insight and familiarity with the various techniques and notions of convergence in the theory of random variables (in probability, almost sure, L^p, in distribution). 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 advanced undergraduate and postgraduate levels.

## Intended Learning Outcomes

Intended learning outcomes

To gain insight and familiarity with the various techniques and notions of convergence in the theory of random variables (in probability, almost sure, L^p, in distribution). 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 advanced undergraduate and postgraduate levels.

## Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## Assessment Information

80% Timed, open-book examination 20% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH30027).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.