Prerequisites

The Mastermath course "Measure-Theoretic Probability" is sufficient. Alternatively: basic knowledge of Probability (equivalent to Chapters 1-8 of "A First Course in Probability" by S. Ross, 9th Edition, or Chapters 1-5 of "Statistical Inference" by G. Casella and R. Berger, 2nd Edition), and of Measure and Integration (equivalent to Chapters 1-5 of "Measure Theory" by D. Cohn, 2nd Edition).

Aim of the course

The aim of this course is to cover the elementary theory of stochastic processes by discussing some of the fundamental classes of processes, namely Brownian motion, continuous-time martingales and Markov and Feller processes. At the end of the course the student: - Is able to recognize the measure-theoretic aspects of the construction of stochastic processes, including the canonical space, the distribution and trajectory of a stochastic process, 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 the fundamental result concerning them, including the optional sampling theorem and Doob's inequality. - Can define Markov processes and continuous-time Markov chains; 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 prove the fundamental Hile-Yosida theorem relating generators and operator semigroups.

Lecturers

Daniel Valesin (RuG), Christian Hirsch (RuG)