Nach positiver Absolvierung der Lehrveranstaltung sind Studierende in der Lage, die grundlegenden Konzepte zu verstehen, die der Theorie zeitdiskreter und zeitkontinuierlicher Markov-Ketten und zeitdiskreter Martingale zugrunde liegen. Sie sind auch in der Lage, diese Konzepte in einer Vielzahl von Anwendungen anzuwenden.

Stochastic processes are mathematical objects aimed at modelling random phenomena evolving in time. We will deal with two of the simplest, and at the same time most important, types of stochastic processes: Markov chains and martingales. We will also see these models at work in a wide variety of applications.

MARKOV CHAINS

• Basic theory

• (Strong) Markov property 
• Communicating classes
• Hitting times
• Recurrence and transience
• Invariant distributions
• Convergence to equilibrium
• Time reversibility
• Ergodic theorem

• Further theory in continuous time

• Generator matrices
• Jump chains and holding times
• Explosion
• Forward and backward equations

• Applications

• Random walks
• Birth-and-death processes
• PageRank algorithm
• Markov chain Monte Carlo (MCMC) methods
• Wright-Fischer model in population genetics
• Poisson processes
• Queuing systems

MARTINGALES

• Basic theory

• Filtrations and adapted processes
• Martingales, supermartingales and submartingales
• Martingale transform
• Stopped martingales
• Doob's optional stopping theorem
• Doob's decomposition
• Maximal inequalities

• Asymptotic theory

• Almost sure convergence: Doob's forward convergence theorem
• Convergence of martingales in L^2
• Uniform integrability and convergence in L^1
• Levy's upward and downward theorems
• Doob's L^p inequality and convergence in L^p

• Applications

• Branching processes
• Games and gambling strategies
• Insurance modelling
• Statistical hypothesis testing
• Filtering problems
• Strong law of large numbers
• Kolmogorov's zero-one law

Vortrag mit Tafel

Präsenzunterricht. Aufgrund des COVID-Infektionsgeschehens kann es im Abhaltemodus zu Änderungen kommen.

Written exam. For those who attend the course, there will be the possibility to collect additional bonus points by answering quick recap questions at the beginning of each lecture. These points will be added to the mark of their written exam (only the first time they take the exam).

If you wish to take the exam after the end of the summer semester 2023, please get in touch directly with the lecturer and another exam will be scheduled.

• Norris, J. (1997). Markov Chains. Cambridge University Press. doi:10.1017/CBO9780511810633
• Williams, D. (1991). Probability with Martingales. Cambridge University Press. doi:10.1017/CBO9780511813658

Grundlegende Wahrscheinlichkeitstheorie, Analysis und lineare Algebra