Students will develop:

Understanding of the modern theoretical and applicable methods and tools of the vast field of Probability Theory (Stochastics) that are at the intersection of many mathematical disciplines.

Recognition of the central part of STOCHASTICS (probability theory, statistics, stochastic processes, stochastic analysis) within almost all science fields and of its explanatory power.

Students will be encouraged to develop:

Ability to read, understand and communicate research literature.

Ability to recognise potential research opportunities and research directions.

(LO1) Detailed understanding of how determinism, in view of complexity, can be handled by random tools.

(LO2) Ability to use probabilistic tools to model, analyse and understand complex systems.

Brief review of prerequisites from discrete probability and related areas.
Sets and measures, their construction, their limits, their meaning.
Random objects, algebras and collections of algebras as mathematical manifestation of information.
Basic background in analysis: derivatives, integrals, spaces of functions, convergence, Fourier.
The fundamental theorems of probability and statistics and limit theorems.
Spaces of measures, convergence and applications: Brownian motion and diffusions.
Elements of the theory of stochastic processes.
Random dynamics, Markovian systems, ergodicity.
Random measures and complete independence.
Case studies of stochastic models from various applications.