Measure Theory and Stochastic Integration

5 credits

Syllabus, Master's level, 1MA051

A revised version of the syllabus is available.

Code: 1MA051
Education cycle: Second cycle
Main field(s) of study and in-depth level: Financial Mathematics A1F, Mathematics A1F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 15 December 2008
Responsible department: Department of Mathematics

Entry requirements

120 credit points including Measure and Integration Theory I

Learning outcomes

In order to pass the course (grade 3) the student should

understand Brownian motion as a stochastic process on a filtered measurable space;

know the class of continuous martingales;

know the construction of a stochastic integral;

know how to use Ito's formula

understand the concept of "quadratic variation" and the martingale characterisation of Brownian motion;

know the representation theorem for martingales and how to use it;

know existence and uniqueness theorems for stochastic differential equations;

be able to use diffusion processes as a tool for mathematical modelling;

understand the connection between diffusion processes and solutions of parabolic and elliptic partial differential equations;

be able to 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

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