Syllabus for Measure Theory and Stochastic Integration

Learning outcomes

In order to pass the course (grade 3) 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.