# Bálint Tóth:

# Martingale Theory (MATH 36204, MATH M6204)

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

__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:

## 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, H****o****lder****, 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**