# 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

## 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

## LECTURE NOTES OF EARLIER VERSIONS OF THIS UNIT:

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

## 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

AVAILABLE NOW!!!

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

AVAILABLE NOW!!!

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

AVAILABLE NOW!!!

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

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)