Martingale Theory with Applications, Autumn 2026

MATH30027, MATHM0045


 
Márton Balázs
Email:m.balazs@our_city.ac.countrycode
Tel:+44 (0) 117 4557958
Office:Fry 1.44
Drop in Sessions: Wednesdays 16:30 - 17:30, 1.44 Fry in teaching weeks.

 
 
  • The unit description for level H/6, including assessment methods, texts, syllabus.
     
  • The unit description for level M/7, including assessment methods, texts, syllabus.
     
  • This unit will happen face-to-face, and you are expected to follow lectures in the classroom. Nevertheless, I will link YouTube videos from previous years on most but not all of the material, see the column "Past videos" in the table. I hope YouTube works for everyone interested, let me know otherwise.
     
  • Calculators are not allowed in the examination, nor are cheat sheets.
     
  • I will assume familiarity with the following concepts from calculus and analysis, but of course I'm happy to discuss these outside class:
    • basic properties, derivatives and integrals of elementary functions like polynomials, exponential, logarithm, trigonometric and some hyperbolic functions
    • the definition of limit, liminf and limsup of real-valued sequences
    • that monotone sequences have limits
    • the definition of a Cauchy sequence and that these are convergent in ℝ
    • that 1/np is summable for p>1 but not for p≤1, and the analogous statement for integrals
    • that exponentials converge/diverge faster than any polynomial
    • the triangle inequality for absolute value
    • the way to find maxima and minima of smooth real to real functions
    • the Heine–Cantor theorem on uniform continuity of continuous real-valued functions on a closed and bounded real interval

     
  • Lecture notes (last modified: 13 Oct 2025). Please send me comments (e.g., typos). Most of these notes are based on A.N. Shyriaev: Probability (Second Edition, Springer) and D. Williams: Probability with Martingales (Cambridge University Press) which you can take a look into yourself. These latter are however advisory, examinable is what is featured in lectures. Accessible, html format.
     
  • 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 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 were actually part of lecture notes for a rather strong first probability unit I used to teach before.)
     
  • Remark on the assessments: there will be no distinction between levels H/6 and M/7 regarding homeworks. Eight homework sets will be assigned, see the schedule below. Please note, these qualify as progress check, meaning no late submissions will be accepted. Our marking deadline is one week after the respective due dates. From each of these, you can collect 10 homework marks, but only the six best of eight will be counted in your coursework mark (and obviously scaled up so that 60 marks count as 100% coursework achievement). Your final mark will be weighted as 20% -- 80% homework mark -- exam mark. Discussions with your fellow students are encouraged. However, you should think about and write up your own solution. Getting external players (friends, AI, tutors) to do this might raise a few marks for you, but more than likely will cost you much more on the exam due to lack of practice that these problems are meant to provide.
     
  • A few more remarks on the exam: for both levels, final examination will be 2½-hour long, will consist of four questions each of which will be used for assessment. The exams at levels H/6 and M/7 will have cca. 80% overlap. Calculators or cheat sheets will not be allowed in the exam. Past exams, one with solutions, are to be found on the Blackboard page Resources for students → Examinations.
     
  • 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.

Homeworks are/will also be posted here: just click those with a link below. They are due on each Thursday of Weeks 3-5 and 7-11 at noon in Blackboard.

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


 
Date Topics Past videos Homework due:
Tue 22 Sep Measure Theory (basic notions, probability) (first year version)
 1, 2 --
Thu 24 Sep Measure Theory (basic notions, probability) (first year version)
 3, 4   --
Fri 25 Sep Sigma-algebras, random variables, stochastic processes 5 --
Tue 29 Sep Random variables, expectation
 6, 7 --
Thu 1 Oct Conditional expectation 8, 9, 10 --
Fri 2 Oct Conditional expectation 11
 --
Tue 6 Oct Conditional expectation --
Thu 8 Oct Probabilistic tools: Borel Cantelli lemmas 12, 13, 14 by noon:
HW1
Fri 9 Oct Probabilistic tools: limits and expectations 15, 16 --
Tue 13 Oct Probabilistic tools: limits and expectations; Fubini 17, 18 --
Thu 15 Oct Probabilistic tools: inequalities 19 by noon:
HW2
Fri 16 Oct Inequalities; modes of convergence 20, 21, 22 --
Tue 20 Oct Modes of convergence: relations 23, 24 --
Thu 22 Oct Filtrations, martingales 25 by noon:
HW3
Fri 23 Oct Stopping times, optional stopping 26, 27, 28 --
Tue 27 Oct Ex. class (probabilistic tools) 29, 30, 31, 32 --
Thu 29 Oct Ex. class (martingales: ABRACADABRA) --
Fri 30 Oct Ex. class (random walks and martingales) --
Tue 3 Nov Upcrossing Lemma, martingale convergence 33, 34 --
Thu 5 Nov Martingale convergence by noon:
HW4
Fri 6 Nov L2 convergence; Doob's decomposition 35
 --
Tue 10 Nov Doob's decomposition --
Thu 12 Nov Uniform integrability 36, 37 by noon:
HW5
Fri 13 Nov UI and convergence: proofs 38, 39, 40, 41, 42 --
Tue 17 Nov UI martingales; tail σ-algebra; 0-1 law 43, 44, 45 --
Thu 19 Nov Strong Law of Large Numbers; Doob's submartingale inequality 46, 47, 48 by noon:
HW6
Fri 20 Nov Doob's submartingale inequality --
Tue 24 Nov Doob's submartingale inequality --
Thu 26 Nov Azuma-Hoeffding concentration by noon:
HW7
Fri 27 Nov Azuma-Hoeffding concentration; European option, hedging strategy 49, 50 --
Tue 1 Dec Black-Scholes optional pricing and proof 51 --
Thu 3 Dec Black-Scholes optional pricing proof 52 by noon:
HW8
Fri 4 Dec Black-Scholes optional pricing proof; (Martingale convergence, UI martingales) --
Tue 8 Dec Ex. class (Martingale convergence, UI martingales) --
Thu 10 Dec Ex. class (Azuma-Hoeffding) --
Fri 11 Dec Q&A --

 
 

 

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