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MA40058: Probability with martingales

Follow this link for further information on academic years Academic Year: 2013/4
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6
Further information on unit levels Level: Masters UG & PG (FHEQ level 7)
Further information on teaching periods Period: Semester 2
Further information on unit assessment Assessment: EX 100%
Further information on supplementary assessment Supplementary Assessment: Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this unit you must take MA40042 or you must take MA30089 and have consulted the unit lecturer.
Further information on descriptions 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. 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.
Further information on programme availabilityProgramme availability:

MA40058 is Optional on the following programmes:

Department of Mathematical Sciences
  • USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
  • USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
  • USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
  • USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
  • USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
  • USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
  • USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
  • TSMA-APM09 : MSc Mathematical Sciences (Part-time)
  • TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time)
  • TSMA-AWM14 : MSc Modern Applications of Mathematics (Full-time incorporating placement)
  • TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time)

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.