This is the course page of MATS254 Martingale theory, lectured in spring 2014 (Course page in Korppi).

Teaching

Matti Vihola <matti.s.vihola at jyu.fi> and Heikki Seppälä <heikki.i.seppala at jyu.fi>

Lectures: 4h weekly for 6 weeks starting January 13

• Mon 12:15–14:00 at MaD380
• Tue 8:15–10:00 at MaD380

Tutorials 2h weekly for 6 weeks starting January 20.

• Mon 10:15–12:00 at MaD380

Lectures

Summary of contents covered:

• Week 1 (13-14.1.): Introduction; conditional expectation & properties
• Week 2 (20-21.1.): Regular conditional distribution; martingales; examples of martingales; basic properties of martingales
• Week 3 (27-28.1.): Stopping times; Doob decomposition; optional stopping theorem
• Week 4 (3-4.2.): Doob's maximal inequalities; Azuma-Hoeffding inequality; uniform integrability
• Week 5 (10-11.2.): Martingale convergence; optional stopping for UI martingales;
• Week 6 (17-18.2.): Kolmogorov's zero-one law; Kakutani's theorem; martingale convergence & Kronecker's lemma; martingale proof of Radon-Nikodym theorem

Tutorials

Tutorial problems sheets:

• Tutorials 1 (20.1.): PDF
• Tutorials 2 (27.1.): PDF
• Tutorials 3 (3.2.): PDF
• Tutorials 4 (10.2.): PDF
• Tutorials 5 (17.2.): PDF
• tutorials 6 (24.2.): PDF

Solutions will be available in Koppa.

Bonus points

By completing exercise problems prior the tutorial sessions, you can accumulate bonus points affecting the final grade. The maximum number of bonus points will increase the final mark by one (at least).

Completed	25%	50%	70%	90%
Points	+2	+3	+4	+5

In case you cannot attend a tutorial session, you can return your solutions to the questions also by email (only PDF files accepted; if scanned hand-written, please write clearly!).

You can give your solutions either in English or in Finnish.

Requirements

In order to pass the course, you need to pass the exam (without the bonus points taken into account). Then, your final mark will be computed from exam points + bonus points.

Supporting material

• S. Geiss: Stochastic processes in discrete time, Lecture notes 62, Department of Mathematics and Statistics, University of Jyväskylä, 2009.
  Available in Kampus Data