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MA40058: Probability with martingales

Follow this link for further information on academic years Academic Year: 2019/0
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
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 Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (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 module 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 Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
  • RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
  • TSMA-AFM17 : MRes Statistical Applied Mathematics
  • TSMA-AFM08 : MSc Modern Applications of Mathematics
  • TSMA-AFM16 : MSc Statistical Applied Mathematics
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AFB05 : BSc(Hons) Statistics (Year 3)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)

Notes: