# Syllabus for Measure Theory and Stochastic Integration

Måtteori och stokastisk integration

A revised version of the syllabus is available.

## 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: The Faculty Board of Science and Technology
• Revised: 2012-04-19
• Revised by: The Faculty Board of Science and Technology
• Applies from: week 30, 2012
• Entry requirements: 120 credits including Integration Theory, 10 credits, or Measure and Integration Theory I, 5 credits
• Responsible department: Department of Mathematics

## Learning outcomes

In order to pass the course (grade 3) the student should
<ul>
<li>understand Brownian motion as a stochastic process on a filtered measurable space;
</li>
<li>know the class of continuous martingales;
</li>
<li>know the construction of a stochastic integral;
</li>
<li>know how to use Ito's formula
</li>
<li>understand the concept of "quadratic variation" and the martingale characterisation of Brownian motion;
</li>
<li>know the representation theorem for martingales and how to use it;
</li>
<li>know existence and uniqueness theorems for stochastic differential equations;
</li>
<li>be able to use diffusion processes as a tool for mathematical modelling;
</li>
<li>understand the connection between diffusion processes and solutions of parabolic and elliptic partial differential equations;
</li>
<li>be able to use Girsanov's representation theorem. </li>
</ul>

## 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

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.