| Unit name | Martingale Theory with Applications 4 |
|---|---|
| Unit code | MATHM6204 |
| Credit points | 10 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Yu |
| Open unit status | Not open |
| Pre-requisites |
Applied Probability 2 (MATH 21400) |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
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. In addition, students will have the opportunity to study the topic of uniformly integrable martingales through directed readings from the book by Williams. Along with lectures given by the main lecturer, there will be one week of lectures given by other staff that are devoted to specific examples of the application of martingale theory in current area of research in probability and statistics.
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.
Syllabus
Conditional expectation; definition of Martingales; optimal stopping theorem; martingale convergence theorem; L^2 martingales; Doob-Meyer decomposition; example of applications given by guest lecturers.
Relation to Other Units
Applied Probability 2 has introduced Martingales, but only covers the most basic of results. This unit will prove most of the results in a rigorous 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 (MATH 35400) and Stochastic Processes (MATH M6006). Compared to the level 3 version of this unit, the level M version has the additional requirement of direct self study based on readings from the book by Williams, thus will have gained a more.
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. Better ability to learn material by directed reading.
Lectures (15) and revision classes; weekly homework assignments to be done by the student and handed in for marking; self-study with directed reading based on the set book and exercises assigned by the lecturer.
The final assessment mark will be based on:
a 1½-hour written examination (80%) in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators will NOT be permitted during this examination. a written essay (20%), involving the independent study and critical evaluation of topics in uniformly integrable martingales, based on directed reading from the book by Williams. Successful completion of the essay will require substantially greater depth of understanding of the area as a whole than that required for the corresponding Level 3 unit.
Williams, D., Probability with Martingales (CUP), Chapters 9-12.
