Syllabus for Probability and Martingales

Sannolikhetsteori och martingaler


  • 10 credits
  • Course code: 1MS045
  • Education cycle: Second cycle
  • Main field(s) of study and in-depth level: Mathematics A1N, Financial Mathematics A1N

    Explanation of codes

    The code indicates the education cycle and in-depth level of the course in relation to other courses within the same main field of study according to the requirements for general degrees:

    First cycle

    • G1N: has only upper-secondary level entry requirements
    • G1F: has less than 60 credits in first-cycle course/s as entry requirements
    • G1E: contains specially designed degree project for Higher Education Diploma
    • G2F: has at least 60 credits in first-cycle course/s as entry requirements
    • G2E: has at least 60 credits in first-cycle course/s as entry requirements, contains degree project for Bachelor of Arts/Bachelor of Science
    • GXX: in-depth level of the course cannot be classified

    Second cycle

    • A1N: has only first-cycle course/s as entry requirements
    • A1F: has second-cycle course/s as entry requirements
    • A1E: contains degree project for Master of Arts/Master of Science (60 credits)
    • A2E: contains degree project for Master of Arts/Master of Science (120 credits)
    • AXX: in-depth level of the course cannot be classified

  • Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
  • Established: 2022-03-03
  • Established by: The Faculty Board of Science and Technology
  • Applies from: Autumn 2022
  • Entry requirements:

    120 credits including 90 credits in mathematics. Probability theory II. Proficiency in English equivalent to the Swedish upper secondary course English 6.

  • Responsible department: Department of Mathematics

Learning outcomes

On completion of the course, the student should be able to:

  • give an account of basic properties of integration on probability spaces,
  • give an account of basic convergence concepts and results,
  • formulate and apply central theorems in probability and martingale theory, and be able to give an account of their proofs,
  • account for financial models in discrete time, and be able to calculate replicating strategies and fair prices of financial claims,
  • use the theory, methods and techniques of the course to solve problems.


Probability spaces, probability measures and random variables. Integration with respect to a probability measure. Different notions of convergence and convergence theorems (monotone, Fatou, dominated). Radon-Nikodym derivatives and conditional expectations. Fair games and martingales, submartingales and supermartingales. Doob decomposition theorem. Stopping times and the optional sampling theorem. The upcrossing inequality and the martingale convergence theorem. The Doob maximal inequality and martingale transforms. Discrete time models in finance. Self-financing portfolios and value processes. Arbitrage opportunities and equivalent martingale measures. Completeness of a market. Options and option pricing.


Lectures and problem solving sessions.


Written examination at the end of the course (6 credits), and assignments during the course (4 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

The course cannot be credited along with the course Probability theory 1MS038.

Reading list

Reading list

Applies from: Autumn 2022

Some titles may be available electronically through the University library.

  • Williams, David Probability with martingales

    New York: Cambridge University Press, 1991

    Find in the library