NMAK24007U Brownian Motion (BM)

Volume 2025/2026
Education

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

Content
  • Existence of a Brownian motion with continuous sample functions
  • Sample path properties of Brownian motion: quadratic variation. non-differentiability, law of the iterated logarithm.
  • Independence and conditional independence.
  • The strong Markov property for Brownian motion.
  • Optional sampling for Brownian motion.
  • Skorokhod embedding.
  • Weak convergence of random walks to Brownian motion.
     
Learning Outcome

Knowledge:

Knowledge of

  • The basic framework of stochastic processes in continuous time.
  • Properties of Brownian motion

 

Skills:

Ability to

  • Transfer statements of independence and conditional independence in a systematic way.
  • 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
Written
Oral
Continuous feedback during the course of the semester
Credit
7.5 ECTS
Type of assessment
Oral examination, 30 minutes (30-minute preparation time)
Examination prerequisites

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

Aid
All aids allowed

All aids allowed during preparation

Marking scale
7-point grading scale
Censorship form
External censorship
Re-exam

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.