# Syllabus for Measure Theory and Stochastic Integration

Måtteori och stokastisk integration

## Syllabus

• 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.

## Content

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.

## Instruction

Lectures and problem solving sessions.

## Assessment

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.

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

Mandatory

• 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