SF2970 Martingales and Stochastic Integrals 6.0 credits

The overall purpose of the course is that the student should be well acquainted with the basic parts of stochastic calculus, including stochastic differential equations and Itô calculus, with applications e.g., to control theory, signal processing and mathematical finance.

Course offerings are missing for current or upcoming semesters.
Headings with content from the Course syllabus SF2970 (Autumn 2007–) are denoted with an asterisk ( )

Content and learning outcomes

Course contents

Discrete and Continuous-time martingales, Wiener process, Stochastic integrals, Itô's lemma, Stochastic differential equations, exponential martingales, Girsanov transformation and its applications, Random time changes.

Intended learning outcomes

To pass the course, the student should be able to do the following:

• Be able to define and account for conditional expectation, filtrations and the martingale property in discrete and continuous time.
• Account for the properties of the Brownian motion (Wiener process), with applications.
• Define and account for Itô's stochastic integrals, the Itô lemma, Girsanov transform, the Martingale Representation Theorem and random time-change of Itô integrals in concrete situations.
• Account for and determine strong and weak solutions of stochastic differential equations of Itô type (diffusion processes).
• Account for and determine stochastic representations of solutions of parabolic partial differential equations (Kolmogorov's forward and backward equations, the Feynman-Kac and Dynkin's formulas).

• Combine all the concepts and methods mentioned above in order to solve more complex problems.

Course disposition

No information inserted

Literature and preparations

Specific prerequisites

SF2940 (5B1540) Probability theory.

Recommended prerequisites

No information inserted

Equipment

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Literature

Djehiche Boualem: Stochastic Calculus, An Introduction with Applications. Compendium from KTH.
Complemental material from the department.

Examination and completion

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

A, B, C, D, E, FX, F

Examination

• TEN1 - Examination, 6.0 credits, grading scale: A, B, C, D, E, FX, 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.

Written examination (6 university credits)

Opportunity to complete the requirements via supplementary examination

No information inserted

Opportunity to raise an approved grade via renewed examination

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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 SF2970

Mathematics

Second cycle