LFU:online

Absolvents should acquire the abilities to understand and express the contents of the course.

This course is about continuous-time stochastic processes and stochastic calculus. Stochastic calculus is the foundation of both mathematical finance and a multitude of models from theoretical physics such as the diffusion of particles.

The course starts with the theory of continuous-time martingales, the first highlight being the celebrated Doob-Meyer decomposition. We shall continue with a detailed analysis of Brownian motion which goes beyond the material covered in the basic course, Introduction to higher stochastics. Further contents of the course are stochastic integration, Ito's formula, the martingale representation theorem, Girsanov's theorem, local times of Brownian motion, and stochastic differential equations.

In the second part of the course we shall investigate the connections between diffusion processes and partial differential equations, most notably, the connection between Brownian motion and the Dirichlet problem.

Lectures, exercise classes, homework.

Assessment is based on the evaluation of homework, presentations given by students in the exercise classes and a written exam.

Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus.

Hui-Hsiung Kuo, Introduction to Stochastic Integration.

Peter Mörters, Yuval Peres, Brownian motion.

Bernt Øksendal, Stochastic differential equations. An introduction with applications.

Philip Protter, Stochastic integration and differential equations. A new approach.

J. Michael Steele, Stochastic Calculus and Financial Applications.

Stochastics 1 and 2; ideally Introduction to higher Stochastics.

Basic knowledge of discrete-time martingales and Brownian motion is required.

The course will be given in English.