MM814: Probability Theory II (5 ECTS)

STADS: 13004601

Level
Master's level course

Teaching period
The course is offered in the spring semester.
4th quarter.

Teacher responsible
Email: njn@imada.sdu.dk

Timetable
There is no timetable available for the chosen semester.

Comment:
Samlæses med MM513, se skema under MM513.

Prerequisites:
None

Academic preconditions:
The students must be familiar with the contents of the courses MM517 Measure- and Integration Theory and MM506 Probability Theory I".

Course introduction
To give the students a solid introduction to the basic theory of stochastic processes in discrete time with special focus on martingales.

Expected learning outcome
By the end for the course, the student will be able to

• reproduce definitions in probability theory within the scope of the course’s syllabus
• reproduce results in probability theory, together with their proofs, within the scope of the course’s syllabus
• apply the theory to solve problems in probability theory
• relate the results within the scope of the course’s syllabus to each other
• explain, with proofs, the different modes of convergence of stochastic processes, and how they relate to each other
• explain, with proofs, the significance of the martingale property of stochastic process, in particular with respect to its convergence properties
• identify martingales and apply the results thereof in probability theory
• predict the long-time behavior of a stochastic process by applying the theory of martingales
• apply martingales to solving problems in analysis, measure theory and finance

Subject overview
1. Modes of convergence of stochastic processes in discrete time.
2. Existence and uniqueness of conditional expectation for integrable stochastic variables.
3. Martingales, optional stoppingtimes and optional sampling.
4. Super- and sub-martingales.
5. The martingale convergence theorem.
6. The Strong Law of Large Numbers.
7. Uniformly integrable martingales.

Literature
There isn't any litterature for the course at the moment.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
(a) A number of mandatory assignments during the course. Pass/fail. Internal evaluation by the teacher. These assignments must be passed in order to be enrolled for the exam.
(b) Oral exam. External examiner, grades according to the 7-point marking scale.

Reexamination after 2nd quarter. The exam form at the re-exam can be different than at the ordinary exam.

Expected working hours
The teaching method is based on three phase model.

Forelæsninger: 28 timer.
Eksaminatorietimer: 18 timer
Educational activities

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.