NMAK11011U Advanced Probability Theory 2 (VidSand2)
Volume 2014/2015
Education
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Actuarial Mathematics
MSc Programme in Statistics
MSc Programme in Actuarial Mathematics
Content
- 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.
Learning Outcome
Knowledge:
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.
Skill:
- 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.
Competence:
- 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.
Academic qualifications
Advances probability
theory 1(VidSand1) or equivalent
Teaching and learning methods
5 hours of lectures and 4
hours of exercises per week for 7 weeks.
Workload
- Category
- Hours
- Exam
- 1
- Lectures
- 35
- Preparation
- 132
- Project work
- 10
- Theory exercises
- 28
- Total
- 206
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Exam
- Credit
- 7,5 ECTS
- Type of assessment
- Oral examination, 30 min30 min preparation. All written aids allowed under preparation and examination.
- Exam registration requirements
- To participate in the exam the compulsory assignment must be approved and valid.
- Aid
- Written aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.
Course information
- Language
- English
- Course code
- NMAK11011U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 2
- Schedule
- C (Mon 13-17 + Wednes 8-17)
- Course capacity
- No limit
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Course responsibles
- Anders Rønn-Nielsen (9-6c7d797470777e70794b786c7f73397680396f76)
Phone +45 35 32 07 17, office
04.3.26
Saved on the
01-12-2014