## Martingale Theory with Applications, Autumn 2018

 Márton Balázs Email: m.balazs@our_city.ac.countrycode Tel: +44 (0)117 928-7991 Office: Maths 3.7 Drop in Sessions: Wednesdays 10:30 - 11:30 on Weeks 1-6, Maths 3.7

 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.   I'm typing up lecture notes as the unit proceeds. Please send me comments (e.g., typos).   Here is last year's unit, and here is how Bálint Tóth taught this unit last time. There are links to various teaching materials from his and even earlier courses. Notice that classes taught in different years are not identical, this current class might differ from previous years' at certain points.   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 has been 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 blue locker with "Martingale Theory" on it on the ground floor of the Main Maths Building.
Homework solutions will appear on Blackboard, please email me if you have problem accessing these.

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

Day Topics Homework due:
Mon 1 Oct Review of the unit; Measure Theory (basic notions, probability) (Sh II.1-II.4; W 1, 2.1-2.4) --
Tue 2 Oct Measure Theory (Sh II.1-II.4; W 1, 2.1-2.4) --
Thu 4 Oct Ex. class (Sigma-algebras, conditional expectation) --
Mon 8 Oct Expectation, conditional expectation (Sh II.6-II.7, W 9.1, 9.2, 9.7) --
Tue 9 Oct Probabilistic tools (Sh II.6, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13) --
Thu 11 Oct Probabilistic tools (Sh II.6, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13)
Modes of convergence, examples (Sh II.10; W 13.5, 13.6, 13.7, A13)
HW1 (sol. on Bb.)
Mon 15 Oct Modes of convergence, examples (Sh II.10; W 13.5, 13.6, 13.7, A13)
--
Tue 16 Oct Uniform integrability (Sh II.6; W 13.1-13.4) --
Thu 18 Oct Ex. class (martingales, optional stopping, ABRACADABRA) (W 10.1-10.4, 10.8-10.11) --
Mon 22 Oct Filtrations, martingales (W 10.1-10.4) --
Tue 23 Oct (Finishing the UI proof;) Optional stopping (W 10.5-10.11) --
Thu 25 Oct Applications: random walks. (W 10.12) --
Mon 29 Oct Upcrossing Lemma, martingale convergence (incl. L2) (W 11.1-11.7, 12.0-12.1) HW2 (sol. on Bb.)
Tue 30 Oct Uniformly integrable martingales (W 14.0-14.2) --
Thu 1 Nov Ex. class (martingales + convergence; 3.10 from sheet) --
Mon 5 Nov Kolmogorov's 0-1 Law; Strong Law of Large Numbers (W 14.3-14.5) --
Tue 6 Nov Strong Law of Large Numbers; Doob's submartingale inequality; Black-Scholes formula (W 14.6, 15.0-15.2) --
Thu 8 Nov Black-Scholes formula (W 15.0-15.2) --
Mon 12 Nov -- Teaching has ended by now -- HW3 (sol. on Bb.)