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-Economics

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 minutes
30 min preparation. All written aids allowed during preparation.
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