Entry requirements

120 credits including 90 credits in mathematics. Integration Theory.

Learning outcomes

To pass the course the student should be able to

• use measure-theoretic and analytic techniques for the derivation of equations describing Markov processes and diffusion processes;
• outline proofs of important theorems of continuous-time martingale processes;
• construct probability measures on infinite dimensional spaces and, in particular, function spaces;
• effectively compute conditional expectations for mutually absolutely continuous stochastic processes and explain the main ideas of the proofs;
• apply stochastic calculus to derive solutions, or properties of solutions, of stochastic differential equations;
• solve partial differential equations by means of Brownian motion and potential theory;
• construct and compute with Poisson processes on Polish spaces and derive Lévy processes and their Wiener-Hopf factorisation;
• effectively use the theory of martingales and compensators to derive thermodynamic limits of stochastic systems;
• give examples of applications of stochastic processes and compute invariant measures.

Content

Brownian motion, continuous-time martingales, Markov processes and their generators, diffusions, stochastic calculus and differential equations, Poisson random measures, point processes, Lévy processes.

Instruction

Lectures and problem solving lessons.

Assessment

The assessment will be based on individual projects at the end of the course. Compulsory assignments will be given during the course.