## Martingale Theory with Applications, Autumn 2017

 Márton Balázs Email: m.balazs@our_city.ac.countrycode Tel: +44 (0)117 928-7991 Office: Maths 3.7 Drop in Sessions: Send me an email to set up an appointment. Q&A Session: Monday 15th January, 16:00, SM4. 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.   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 in class, or in the blue locker with my name 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 25 Sep Review of the unit; Measure Theory (basic notions, probability) (Sh II.1-II.4; W 1, 2.1-2.4) --
Tue 26 Sep Measure Theory (Sh II.1-II.4; W 1, 2.1-2.4) --
Fri 29 Sep Ex. class (Sigma-algebras, conditional expectation) --
Mon 2 Oct Expectation, conditional expectation (Sh II.6-II.7, W 9.1, 9.2, 9.7) --
Tue 3 Oct Probabilistic tools (Sh II.6, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13) --
Fri 6 Oct Types of convergence, examples (Sh II.10; W 13.5, 13.6, 13.7, A13) HW1 (sol. on Bb.)
Mon 9 Oct Uniform integrability (Sh II.6; W 13.1-13.4) --
Tue 10 Oct Uniform integrability (Sh II.6; W 13.1-13.4) --
Fri 13 Oct Ex. class (martingales, optional stopping, ABRACADABRA) (W 10.1-10.4, 10.8-10.11) --
Mon 16 Oct Filtrations, martingales (W 10.1-10.4) --
Tue 17 Oct Optional stopping (W 10.5-10.11) --
Fri 20 Oct Applications: random walks. Upcrossing Lemma (W 10.12, 11,1-11.4) HW2 (sol. on Bb.)
Mon 23 Oct Martingale convergence (incl. L2) (W 11.5-11.7, 12.0-12.1) --
Tue 24 Oct Uniformly integrable martingales (W 14.0-14.2) --
Fri 27 Oct Ex. class (martingales + convergence; 3.10 and 3.12 from sheet) --
Mon 30 Oct Kolmogorov's 0-1 Law; Strong Law of Large Numbers (W 14.3-14.5) --
Tue 31 Oct Doob's submartingale inequality; Black-Scholes formula (W 14.6, 15.0-15.2) --
Fri 3 Nov Black-Scholes formula (W 15.0-15.2) HW3 (notes for 3.9(e)) (sol. on Bb.)