NMAK24007U Brownian Motion (BM)

Volume 2024/2025

MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject
MSc Programme in Mathematics-Economics

  • Weak convergence of probability measures. Characteristic functions.
  • The central limit theorem. Triangular arrays and Lindeberg condition. The multivariate central limit theorem.
  • Existence of a Brownian motion with continuous sample functions
  • Sample path properties of Brownian motion: quadratic variation. non-differentiability, law of the iterated logarithm
  • The strong Markov property for Brownian motion
  • Optional sampling for Brownian motion
  • Skorokhod embedding
  • Weak convergence of random walks to Brownian motion
Learning Outcome


  • The fundamental role of Gaussian distributions and its rooting in CLT (central limit theorem)
  • The basic framework of stochastic processes in continuous time
  • Properties of Brownian motion


Skills: Ability to

  • Establish weak convergence results in finite dimension and in separable metric spaces
  • Derive explicit properties of Brownian motion using a variety of methods


Competencies: Ability to

  • Produce independent proofs in extension of the acquired knowledge
Sand2 - alternatively VidSand1 (or Stok2) from previous years

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 3 hours of exercise class for 7 weeks.
  • Category
  • Hours
  • Lectures
  • 35
  • Preparation
  • 149
  • Theory exercises
  • 21
  • Exam
  • 1
  • Total
  • 206
Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Oral examination, 30 minutes (30-minute preparation time)
Exam registration requirements

To participate in the exam the compulsory assignment must be approved and valid

All aids allowed

All aids allowed during preparation

Marking scale
7-point grading scale
Censorship form
External censorship

Same as ordinary exam.

If the compulsory assignment was not approved before the ordinary exam it must be (re)submitted and approved. It must be (re)submitted at the latest three weeks before the beginning of the re-exam week.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.