MM513: Probability theory II (5 ECTS)

STADS: 13001301

Level
Bachelor course

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

Teacher responsible
Email: njn@imada.sdu.dk

Timetable
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Comment:
Ubegrænset deltagerantal. 4. kvartal.
Samlæses med MM814

Prerequisites:
None

Academic preconditions:
The material from the courses MM517 "Measure- and integration theory" and MM506 "Probability theory I" is assumed to be known.

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

Qualifications
The theory for stochastic processes plays an important role in probability theory and its applications in statistics, mathematical finance, functional analysis, etc. The theory of martingales, in particularly, plays a crucial role in the development of stochastic integration as the main tool for solving stochastic differential equations. Having completed the course successfully, the students can be expected to • be familiar with the main types of convergence for stochastic processes in discrete time. • have a solid understanding of the theory of martingales and its applications e.g. to mathematical finance. • be prepared for advanced studies in stochastic differential equations, mathematical finance, statistics and functional analysis.

Expected learning outcome
By the end of 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 a stochastic process, in particular with respect to its convergence properties
• identify martingales and apply the results thereof in probability theory

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

• Meddeles ved kursets start..

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 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 2. 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.

(a) Forelæsninger (28 timer).
(b) Eksaminatorier (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.