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| Unit name |
Brownian Motion |
| Unit code |
MATHM0026 |
| Credit points |
20 |
| 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 |
Analysis 1B (MATH10006), Probability 2 (MATH20008) and ideally but not essentially, at least one of: Further Topics in Probability (MATH30006), Measure Theory and Integration (MATH30007) or Martingale Theory with Applications (MATH36204). Some background in PDE's would be helpful but not essential.
|
| Co-requisites |
None
|
| School/department |
School of Mathematics |
| Faculty |
Faculty of Science |
Description including Unit Aims
The unit aims to give a rigorous yet non-technical introduction to Brownian motion, with an emphasis on concrete calculations and examples. Brownian motion is named after the botanist Robert Brown who, in 1827, observed the apparently random motion of pollen particles under a microscope. In 1905, Albert Einstein argued that this motion is related to the propagation of heat, and hence has a precise mathematical description. This remarkable connection between probability and mathematical physics laid the foundations for many further developments which continue to this day. This unit will introduce and develop the mathematical theory of Brownian motion, as well as some of the main aspects and tools of the study of continuous stochastic processes, which include - Markov processes, Martingales, Ito calculus, local times, SDEs and diffusion processes.
Intended Learning Outcomes
- To gain profound understanding of the basic notions and techniques of the theory of Brownian motion, diffusion processes and applications.
- To prepare the postgraduate student for independent research in mathematics.
- Familiarity with some of the main ideas and examples in the subject.
- Understanding some important theorems and identities, and their proofs.
- Ability to perform some typical computations.
Teaching Information
Lectures, regular formative problem sheets and office hours
Assessment Information
100% examination (2.5 hours)
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.
Reading and References
Recommended:
- K.L. Chung, R. Williams: Introduction to stochastic integration. Second edition. Birkauser, 1989
- I. Karatzas, S. Shreve: Brownian Motion and Stochastic Calculus, Springer 1991
- J. Lamperti, Stochastic Processes: a Survey of the Mathematical Theory, Springer 1977
- B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer 2010
