Bálint Tóth:

Martingale Theory (MATH 36204, MATH M6204)

Autumn 2015, TB1, weeks 1-6 (28 Sep - 08 Nov 2015)

GENERAL COURSE DESCRIPTION FOR MATH 36204

GENERAL COURSE DESCRIPTION FOR MATH M6204

TIMETABLE:

LECTURES:

Wed:   12:00-12:50,   MATH SM3
Fri:      12:00-12:50,   MATH SM4

Fri:      14:00-14:50,   MATH SM4

OFFICE HOUR:

Wed:   14:30-15:30,   MATH 3.12

PRELIMINARY SCHEDULE:

Preliminary schedule of the course

LECTURE NOTES (hand written, downloadable, these notes cover all material done in class):

1. Measure, integration, conditional expectation (crash course) 1: σ-algebra, measurability, measure, Lebesgue integral

2. Measure, integration, conditional expectation (crash course) 2: limits of integrals (Fatou, Beppo Levi, Lebesgue), convexity (Jensen, Holder, Minkowski), Lebesgue spaces

3. Measure, integration, conditional expectation (crash course) 3: singularity, absolute continuity, Radon-Nikodym, Lebesgue decomposition, conditional expectation

4. Filtrations, martingales: definitions and examples, Polya Urn, Galton-Watson branching, gambling and discrete time stochastic integration

5. Stopping times and Optional Stopping Theorem

6. Martingale Convergence Theorem

7. Submartingale maximal inequalities (Doob’s inequalities)

LECTURE NOTES OF EARLIER VERSIONS OF THIS UNIT:

Dr Feng Yu’s lecture notes (2010)

Dr Nic Freeman’s lecture notes (2015, spring)

BOOKS, ADDITIONAL READING: There are many excellent books on the subject of this course. Our favourite is:

D. Williams: Probability with Martingales. Cambridge University Press, 1991

PREREQUESITES:

Probability 1&2

Fluency in real analysis

PROBLEM SETS, HOME WORK ASSIGNMENTS:

(Solutions will be available after hw due date)

1. Measure and integration: HW: 1.3, 1.5, 1.9; Bonus: 1.7, 1.8, 1.12; Due: 12 Oct

Solutions AVAILABLE NOW!!!

2. Conditional expectation: HW: 2.3, 2.4, 2.5; Bonus: 2.6; Due: 19 Oct

Solutions AVAILABLE NOW!!!

3. Filtrations, martingales: HW: 3.3, 3.6, 3.11; Bonus: 3.9, 3.12, 3.15; Due: 26 Oct

Solutions AVAILABLE NOW!!!

4. Optional stopping: HW: 4.7, 4.9, 4.12; Bonus: 4.10, 4.11; Due: 2 Nov

Solutions AVAILABLE NOW!!!

Exam questions and solutions from 2016. January AVAILABLE NOW!!!

FURTHER PROBLEMS FOR SOLUTION:

Here below you will find the homework assignments and exam questions from two earlier years along with their solutions. Note that the unit was taught by other colleagues. So, notation and terminology slightly differs from mine. Also, the material covered differs in cca 15% (I did not cover uniform integrability, but did instead Doob’s (submartingale, maximal) inequality). These problem & solution sets are provided for your convenience only.

à Homework assignments and solutions from 2010 autumn (by Dr Feng Yu)

à Homework assignments and solutions from 2015 spring (by Dr Nic Freeman)

à Exam questions and solutions from 2011 April

à Exam questions from 2015 May