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>
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.
Lectures and problem solving sessions.
Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.