MA40058: Probability with martingales
[Page last updated: 05 August 2021]
Academic Year:  2021/2 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
 Semester 2

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module 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 Economics
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 3)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
 RSMAAFM16 : Integrated PhD Statistical Applied Mathematics
 TSMAAFM17 : MRes Statistical Applied Mathematics
 TSMAAFM16 : MSc Statistical Applied Mathematics
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
 USMAAFB05 : BSc(Hons) Statistics (Year 3)
 USMAAAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
 USMAAKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAFM14 : MMath(Hons) Mathematics (Year 4)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)

Notes:  This unit catalogue is applicable for the 2021/22 academic year only. Students continuing their studies into 2022/23 and beyond should not assume that this unit will be available in future years in the format displayed here for 2021/22.
 Programmes and units are subject to change 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 prerequisite rules.
 Find out more about these and other important University terms and conditions here.
