
Academic Year:  2013/4 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
Semester 2 
Assessment:  EX 100% 
Supplementary Assessment: 
Likeforlike reassessment (where allowed by programme regulations) 
Requisites:  Before taking this unit you must take MA40042 or you must take MA30089 and have consulted the unit lecturer. 
Description:  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. OptionalStopping Theorem. Martingale Convergence Theorem. L^{2} bounded martingales. Doob decomposition. Anglebrackets process. Lévy's extension of the BorelCantelli lemmas. Uniform integrability. UI martingales. Lévy's 'Upward' and 'Downward' Theorems. Kolmogorov 01 law. Martingale proof of the Strong Law. Doob's Submartingale Inequality. Law of iterated logarithm. Doob's L^{p} inequality. Likelihood ratio. Kakutani's theorem. Other applications. 
Programme availability: 
MA40058 is Optional on the following programmes:Department of Mathematical Sciences
