Martingale Theory with Applications, Autumn 2019


 
Márton Balázs
Email:m.balazs@our_city.ac.countrycode
Tel:+44 (0) 117 4284918
Office:Fry 1.44
Drop in Sessions: Thursdays 10:30 - 11:30 on Weeks 1-6, Fry 1.44
Q&A Session: Tuesday 14th January, 9:00am, LG.20. Prepare with questions.

 
 
  • The unit description for level H/6, including assessment methods, texts, syllabus.
     
  • The unit description for level M/7, including assessment methods, texts, syllabus.
     
  • Calculators are not allowed in the examination.
     
  • Remark on the assessments: there will be no distinction between levels H/6 and M/7 regarding homeworks. Three homework sets will be assigned, see the schedule below. My marking deadline is the class one week after the respective due dates. From each of these, you can collect 17 homework marks. Your final mark will be weighted as 20% -- 80% homework mark -- exam mark.
     
  • A few more remarks on the exam: for both levels, final examination will be 1½-hour long, will consist of four questions each of which will be used for assessment. The difference between levels H/6 and M/7 will only be the required depth of reproducing proofs in the exam. On H/6 I can ask for at most a couple of words about how a proof goes. On M/7, I can ask for proofs in details.
     
  • Lecture notes (greener version, half the paper). Please send me comments (e.g., typos).
     
  • Extended slides/notes of elementary probability. Some of it can be used as reference and refreshment for parts of the Probability 1 and Advanced Probability 2 units. We will not touch most of this material, only cover a bit of the measure theoretic foundations in class. Please notice that these links are by no means meant to fully cover our material, nor will all parts of all of them be assessed. They rather serve as background reading. (They are actually part of lecture notes for a rather strong first probability unit I used to teach before.)

Below is a detailed schedule. Topics of future events are plans, and can change. Topics of past events serve as log. The relevant section number from A.N. Shyriaev: Probability (Second Edition, Springer) (Sh) and D. Williams: Probability with Martingales (Cambridge University Press) (W) have also been added. These are advisory, examinable is what is featured on lectures.
Homeworks are/will also be posted here: just click those with a link below. They are due at 12:00pm in the locker with "Martingale Theory" on it in G.90 Fry Building.
Homework solutions will appear on Blackboard, please email me if you have problem accessing these.

I came across this illustration of Jensen's inequality: square-root of the average ≠ average of the square-root.


 
Day Topics Homework due:
Mon 30 Sep Review of the unit; Measure Theory (basic notions, probability) (Sh II.1-II.4; W 1, 2.1-2.4) --
Tue 1 Oct Measure Theory (Sh II.1-II.4; W 1, 2.1-2.4) --
Fri 4 Oct Ex. class (Sigma-algebras, conditional expectation) --
Mon 7 Oct Conditional expectation (Sh II.6-II.7, W 9.1, 9.2, 9.7) --
Tue 8 Oct Expectation, Probabilistic tools (Sh II.6, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13) --
Fri 11 Oct Probabilistic tools (Sh II.6, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13) --
Mon 14 Oct Modes of convergence, examples (Sh II.10; W 13.5, 13.6, 13.7, A13)
 HW1 HW1 (sol. on Bb.)
Tue 15 Oct Uniform integrability (Sh II.6; W 13.1-13.4) --
Fri 18 Oct Ex. class (martingales, optional stopping) (W 10.1-10.4, 10.8-10.11) --
Mon 21 Oct (ABRACADABRA and finishing the UI proof;) Filtrations, martingales (W 10.1-10.4) --
Tue 22 Oct (still that UI proof;) Optional stopping (W 10.5-10.11) --
Fri 25 Oct Applications: random walks. (W 10.12) --
Mon 28 Oct Upcrossing Lemma, martingale convergence (W 11.1-11.7) HW2 HW2 (sol. on Bb.)
Tue 29 Oct L2 convergence; uniformly integrable martingales (W 12.0-12.1, 14.0-14.2) --
Fri 1 Nov Ex. class (martingales + convergence) --
Mon 4 Nov Kolmogorov's 0-1 Law; Strong Law of Large Numbers (W 14.3-14.5) --
Tue 5 Nov Strong Law of Large Numbers; Doob's submartingale inequality; Black-Scholes formula (W 14.6, 15.0-15.2) --
Fri 8 Nov Black-Scholes formula (W 15.0-15.2) --
Mon 11 Nov -- Teaching has ended by now -- HW3 HW3 (notes for 3.9(e)) (sol. on Bb.)

 
 

 

If you have any questions, please contact me (see on top). Click here to see my schedule.


 

To the teaching page