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.