Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Stochastic Processes 3

Practical information
Study year 2016/2017
Block 2
Programme level Bachelor
Course responsible
  • Anders Rønn-Nielsen (9-6576726d6970776972447165786c326f7932686f)
Phone +45 35 32 07 17, office 04.3.26
  • Department of Mathematical Sciences
Course number: NMAB15026U

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.

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.
  • understand 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.
  • describe 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.

Recommended prerequisites

Stochastic Processes (Stok 2) or equivalent

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BSc Programme in Actuarial Mathematics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


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Teaching and learning methods

5 hours of lectures and 4 hours of exercises per week for 7 weeks.


No limit




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


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

Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

Censorship form

External censorship


As the ordinary exam. If the compulsory assignment was not approved before the ordinary exam it must be resubmitted at the latest two weeks before the beginning of the re-exam week. It must be approved before the re-exam

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