Prerequisites
A first basic course in probability theory as treated e.g. in the book by Grimmett and Welsh, "Probability: an introduction." Oxford University Press, 2nd edition 2014. In particular, knowledge in the following topics: probability spaces, conditional probabilities, discrete and continuous real-valued random variables, moments and covariances, law of large numbers, central limit theorem. Further, familiarity with basic key concepts from measure theory such as measurable spaces and functions and main convergence theorems (Fatou, monotone and dominated convergence).

Aim of the course
The aim of the course is to cover the basic theory of stochastic processes via an in-depth description of some fundamental examples, namely, Brownian motion, continuous-time martingales, and Markov and Feller processes.

At the end of the course the student:

• Is able to recognise the measure-theoretic aspects of the construction of stochastic processes, including the canonical space, the distribution and the trajectory, filtrations and stopping times.
• Is able to state Kolmogorov's Extension Theorem and use it to show the existence of stochastic processes with a prescribed distribution.
• Can describe the abstract construction of Brownian motion and prove some of the elementary properties of this process, such as continuity of trajectories and the distribution of the maximum of a trajectory.
• Can define continuous-time martingales (and super- and sub-martingales) and prove some of their fundamental properties, including the optional sampling theorem and Doob's inequality.
• Can define Markov processes and continuous-time Markov chains and, for the latter, can explain the relation between the transition function and the infinitesimal generator, and give a construction involving Poisson processes.
• Can define Feller processes and understand the use of the fundamental Hille-Yosida theorem relating generators and operator semigroups.

Lecturers / Assistant
Evgeny Verbitskiy (UL)
Frank den Hollander (UL).

Teaching assistant: Rangel Baldasso (UL).