## 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 Q&A Session: Monday 21st January, 14:00, SM4. Prepare with questions.

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)
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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.)

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

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