Martingale Theory with Applications, Autumn 2020


 
Márton Balázs
Email:m.balazs@our_city.ac.countrycode
Tel:+44 (0) 117 4284918
Office:Fry 1.44 (but I think you are not allowed up here...)
Math cafés: Mondays 11:00am - 12:00pm on Weeks 2-6, Zoom
Drop in Sessions: Wednesdays 12:00pm - 13:00pm on Weeks 1-6, Zoom.
Zoom meeting links will be announed on Blackboard and emailed to you a few days in advance.
Q&A Session: Monday 25th January, 2:00pm, see your email or Blackboard for the Zoom link. Prepare with questions.

 
 
  • The slides from the intro Zoom session.
     
  • The unit description for level H/6, including assessment methods, texts, syllabus.
     
  • The unit description for level M/7, including assessment methods, texts, syllabus.
     
  • 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 three questions each of which will be used for assessment. The exams at levels H/6 and M/7 will have cca. 80% overlap. Past exams, one with solutions, are to be found on the Blackboard page Resources for students → Examinations.
     
  • 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.)
     
  • I came across this illustration of Jensen's inequality: square-root of the average ≠ average of the square-root.

Below is a detailed schedule. Topics of future events are plans, and can change. Topics of past events serve as log. Blended teaching will manifest in prerecorded videos and live, face-to-face teaching. However, I will also broadcast these latter from the classroom via Zoom. Videos can be found under the links below, and I will also back them up on Blackboard. The Zoom meeting links will be announced at least a few days in advance on Blackboard and by email. The relevant section number from the recommended 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 in lectures.

Homeworks are/will also be posted here: just click those with a link below. They are due every second Thursday (see below) at 12:00pm in Blackboard.

Homework solutions will appear on Blackboard, please email me if you have problem accessing these.


 
Time Where Topics Watch prerecorded: Homework due:
Mon 5 Oct, 10:00am Zoom Live session (intorduction and review of the unit) --
by Thu 8 Oct At home Measure Theory (basic notions, probability)
(Sh II.1-II.4; W 1, 2.1-2.4) 		L1, L2, L3, L4, L5
  		--
Fri 9 Oct, 10:00am Zoom+Fry G.10 Live ex.class (sigma-algebras, conditional expectation) --
by Thu 15 Oct At home Expectation, conditional expectation;
probabilistic tools
(most from Further T. in Probability, we only need the times as indicated)
(Sh II.6-II.7, II.10; W 2.7, 6.6, 6.7, 6.8, 6.13, 9.1, 9.2, 9.7) 		L6, L7, L8, L9, L10
F1, F2, F3, F4[-5:51], F5[-5:45]
F6[-2:47], L11, F7[-2:25; 8:17-11:13]
F8[-3:32], F9[-2:52] 		by Thu 15 Oct, noon:
HW1 (sol. on Bb.)
Fri 16 Oct, 10:00am Zoom+Fry G.10 Live session (conditional expectation, probabilistic tools) --
by Thu 22 Oct At home Modes of convergence;
(from Further T. in Probability, we only need the times as indicated)
filtrations, martingales;
Optional stopping
(Sh II.10; W 13.5, 13.6, 13.7, A13; 10.1-10.11)
 F10, F11[-2:21], F12[-1:35]
 
L12
L13, L14, L15
  		--
Fri 23 Oct, 10:00am Zoom+Fry G.10 Live ex.class (martingales, optional stopping) (W 10.1-10.4, 10.8-10.11) --
by Thu 29 Oct At home Upcrossing Lemma, martingale convergence;
L2 convergence
Uniform integrability;
(W 11.1-11.7, 12.0-12.1; Sh II.6) 		L16, L17
L18
L19, L20, L21
  		by Thu 29 Oct, noon:
HW2 (sol. on Bb.)
Fri 30 Oct, 10:00am Zoom+Fry G.10 Live session (applications: random walks. (W 10.12)) --
by Thu 5 Nov At home Uniform integrability;
uniformly integrable martingales;
Kolmogorov's 0-1 Law
(Sh II.6; W 13.1-13.4; 14.0-14.3) 		L22, L23, L24, L25
L26, L27
L28
  		--
Fri 6 Nov, 10:00am Zoom+Fry G.10 Live ex.class (martingales + convergence) --
by Thu 12 Nov At home Strong Law of Large Numbers;
Doob's submartingale inequality;
Black-Scholes formula
(W 14.4-14.5; 14.6, 15.0-15.2) 		L29, L30
L31
L32, L33, L34, L35
  		--
Fri 13 Nov, 10:00am Zoom+Fry G.10 Live session (Black-Scholes formula (W 15.0-15.2)) --
-- -- -- -- by Thu 19 Nov, noon:
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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