Syllabus for Measure Theory and Stochastic Integration

Måtteori och stokastisk integration


  • 5 credits
  • Course code: 1MA051
  • Education cycle: Second cycle
  • Main field(s) of study and in-depth level: Mathematics A1F, Financial Mathematics A1F
  • Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
  • Established: 2007-03-15
  • Established by:
  • Revised: 2019-10-24
  • Revised by: The Faculty Board of Science and Technology
  • Applies from: Autumn 2020
  • Entry requirements:

    120 credits including Integration Theory, 10 credits. Integration Theory may be studied at the same time as 1MA051. 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:

  • interpret Brownian motion as a stochastic process on a filtered measurable space;
  • describe the class of continuous martingales;
  • describe the construction of a stochastic integral;
  • use Ito's formula;
  • describe the concept of "quadratic variation" and the martingale characterisation of Brownian motion;
  • formulate the representation theorem for martingales and how to use it;
  • formulate the existence and uniqueness theorems for stochastic differential equations;
  • use diffusion processes as a tool for mathematical modelling;
  • explain the connection between diffusion processes and solutions of parabolic and elliptic partial differential equations;
  • use Girsanov's representation theorem.


Brownian motion. Stochastic integration. Ito's formula. Continuous martingales. The representation theorem for martingales. Stochastic differential equations. Diffusion processes. Girsanov's representation theorem. Applications from selected areas.


Lectures and problem solving sessions.


Compulsory assignments during the course.

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.

Reading list

Reading list

Applies from: Spring 2021

Some titles may be available electronically through the University library.

  • Øksendal, Bernt Stochastic differential equations : an introduction with applications

    6. ed.: Berlin: Springer, 2003

    Find in the library


  • Revuz, Daniel; Yor, Marc Continuous martingales and Brownian motion

    3. ed.: Berlin: Springer, cop. 1999

    Find in the library

  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus

    2. ed.: Berlin: Springer, cop. 1991

    Find in the library

Reading list revisions

Last modified: 2022-04-26