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FSF3713 Stochastic Analysis 7.5 credits

The course treats the basic theory of stochastic calculus and is aimed towards doctoral students of mathematics or applied mathematics with previous knowledge in measure theory and probability theory.

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Headings with content from the Course syllabus FSF3713 (Spring 2019–) are denoted with an asterisk ( )

Content and learning outcomes

Course contents

  • Stochastic processes, martingales, local martingales, stopping times, filtrations, Markov properties.
  • Brownian motion.
  • The Ito isometry, Ito integrals, Ito’s formula.
  • Existence and uniqueness of solutions to stochastic differential equations.
  • Diffusion processes.
  • Girsanov’s theorem.
  • Probabilistic representations of solutions to partial differential equations.
  • The Feynman­Kac formula, Kolmogorov’s forward and backward equations, recurrence, invariant densities.
  • More advanced topics, e.g., local times, if time permits.

Intended learning outcomes

  • Understand and explain the following concepts: filtration, stochastic process in continuous time, local martingale, martingale, stopping time, quadratic variation.
  • Sketch at least one construction of Brownian motion.
  • Construct the Ito integral in some generality, and discuss its basic properties.
  • Explain Ito’s formula and use it for practical computations of, e.g., Ito integrals.
  • Discuss basic properties of stochastic differential equations (SDEs), in particular diffusions in one dimension.
  • Explain Girsanov’s theorem.
  • Discuss connections between the theory of SDEs and partial differential equations.
  • Solve problems and discuss current research connected to the theory presented in the course

Course disposition

No information inserted

Literature and preparations

Specific prerequisites

Basic probability theory (e.g., SF3940) and basic knowledge in analysis and linear algebra, especially measure theory and Lebesgue integration.

Recommended prerequisites

No information inserted


No information inserted


For example

Karatzas­Shreve “Brownian motion and stochastic calculus” (ISBN 978-1-4612-­0949-­2),

Revuz­Yor “Continuous martingales and Brownian motion” (ISBN 978­-3-662­-06400-­9);

Öksendal “Stochastic Differential Equations” (ISBN 978-­3-­642­14394­-6)

Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

Grading scale

P, F


  • INL1 - Assignment, 3.5 credits, grading scale: P, F
  • TENM - Oral exam, 4.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homework assignments and oral/written exam.

Other requirements for final grade

Homework assignments completed and oral/written exam passed.

Opportunity to complete the requirements via supplementary examination

No information inserted

Opportunity to raise an approved grade via renewed examination

No information inserted


Ethical approach

  • All members of a group are responsible for the group's work.
  • In any assessment, every student shall honestly disclose any help received and sources used.
  • In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

Further information

Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSF3713

Offered by

Main field of study

This course does not belong to any Main field of study.

Education cycle

Third cycle

Add-on studies

No information inserted


Fredrik Viklund (

Supplementary information

PhD students only.

Postgraduate course

Postgraduate courses at SCI/Mathematics