NMAK11011U Advanced Probability Theory 2 (VidSand2)

Volume 2013/2014
Education
MSc programme in Mathematics
MSc programme in Statistics
MSc programme in Acturial Mathematics
MSc programme in Mathematics-Economics
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.
Advances probability theory 1(VidSand1) or equivalent
5 hours of lectures and 4 hours of exercises per week.
  • Category
  • Hours
  • Exam
  • 1
  • Lectures
  • 35
  • Preparation
  • 132
  • Project work
  • 10
  • Theory exercises
  • 28
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Oral examination, 30 min
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Exam registration requirements
To participate in the exam the compulsory assignment must be approved and valid.
Aid
All 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.