In order to pass the course (grade 3) the student should be able to
give an account of important concepts and definitions in the area of the course;
exemplify and interpret important concepts in specific cases;
formulate important results and theorems covered by the course;
describe the main features of the proofs of important theorems;
express problems from relevant areas of applications in a mathematical form suitable for further analysis;
use the theory, methods and techniques of the course to solve mathematical problems;
present mathematical arguments to others.
Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.
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.