Advanced Probability Theory 2 (VidSand2)
- Signed measures, absolute continuity and singularity of measures, the Radon-Nikodym Theorem.
- Conditional expectations given a sigma-algebra.
- Martingales and submartingales in discrete time, the martingale convergence theorem, stopping times and optional sampling.
- Central Limit Theorem for martingales.
- Brownian motion: definition, continuity, variation and quadratic variation, non-differentiability, the law of the iterated logarithm.
MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics
MSc Programme in Mathematics-Economics
MSc Programme in Statistics
Basic knowledge of the topics covered by the course: Decompositions of signed measures, conditional expectations, martingale theory, CLT for martingales, and definition, existence and path properties of the Brownian motion.
- describe and prove the results on decomposition of signed measures.
- use the calculation rules for conditional expectations.
- show whether a sequence of random variables is a martingale or a submartingale.
- derive and describe the main results on martingales.
- apply the results on martingales to concrete examples.
- describe the foundation for the construction of stochastic processes in continuous time.
- describe the basic properties of the sample paths for Brownian motion.
- discuss the relation between decomposition of measures and conditional expectations.
- relate and compare the results on martingales.
- use martingale CLT and understand the result compared to the classical CLT.
- discuss the concept of sample paths with a view to constructing continuous stochastic processes.
- Give an oral presentation of a specific topic within the theory covered by the course.
5 hours of lectures and 4 hours of exercises per week for 7 weeks.
Advances probability theory 1(VidSand1) or equivalent
Academic qualifications equivalent to a BSc degree is recommended.
The course is equivalent to Stochastic Processes 3 (NMAB15026U)
Written feedback in the form of comments to the compulsory
Oral feedback during exercise classes, as a response to the
contribution of the students to the solution process of the
Detailed oral feedback after the oral exam.
- 7,5 ECTS
- Type of assessment
Oral examination, 30 minutes30 min preparation. All aids allowed during preparation.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
Criteria for exam assessment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.
Single subject courses (day)
- Theory exercises
- Project work
- Course number
- 7,5 ECTS
- Programme level
- Full Degree Master
- Block 2
- No limit
- Study Board of Mathematics and Computer Science
- Department of Mathematical Sciences
- Faculty of Science
- Ernst Hansen (8-68756b6471766871437064776b316e7831676e)
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Courseinformation of students