Advanced Probability Theory 2 (VidSand2)

Course 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.

MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics-Economy

Learning outcome


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

7,5 ECTS
Type of assessment
Oral examination, 30 min
30 min preparation. All written aids allowed under preparation and examination.
Written 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)

  • Category
  • Hours
  • Lectures
  • 35
  • Theory exercises
  • 28
  • Exam
  • 1
  • Project work
  • 10
  • Preparation
  • 132
  • English
  • 206