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![]() | 2015/6 |
![]() | Department of Mathematical Sciences |
![]() | 6 |
![]() | Masters UG & PG (FHEQ level 7) |
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Semester 2 |
![]() | EX 100% |
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MA40058 Mandatory extra work (where allowed by programme regulations) |
![]() | Before taking this module you must take MA40042 or you must take MA30089 and have consulted the unit lecturer. |
![]() | Aims: To stimulate through theory and especially examples, an interest and appreciation of the power and elegance of martingales in analysis and probability. To demonstrate the application of martingales in a variety of contexts, including their use in proving some classical results of probability theory. Learning Outcomes: On completing the course, students should be able to: * demonstrate a good knowledge and understanding of the main results and techniques of discrete time martingale theory; * apply martingales in proving some important results from classical probability theory; * recognise and apply martingales in solving a variety of more elementary problems. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials). Content: Review of measure theory; fundamental concepts and results. Conditional expectation. Filtrations. Martingales. Stopping times. Optional-Stopping Theorem. Martingale Convergence Theorem. L2 -bounded martingales. Doob decomposition. Angle-brackets process. Lévy's extension of the Borel-Cantelli lemmas. Uniform integrability. UI martingales. Lévy's 'Upward' and 'Downward' Theorems. Kolmogorov 0-1 law. Martingale proof of the Strong Law. Doob's Submartingale Inequality. Law of iterated logarithm. Doob's Lp inequality. Likelihood ratio. Kakutani's theorem. Other applications. |
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MA40058 is Optional on the following programmes:Department of Mathematical Sciences
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