105.746 Theorie stochastischer Prozesse
Diese Lehrveranstaltung ist in allen zugeordneten Curricula Teil der STEOP.
Diese Lehrveranstaltung ist in mindestens einem zugeordneten Curriculum Teil der STEOP.

2024S, VO, 3.0h, 4.5EC


  • Semesterwochenstunden: 3.0
  • ECTS: 4.5
  • Typ: VO Vorlesung
  • Format der Abhaltung: Präsenz


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.

Inhalt der Lehrveranstaltung

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.


  • 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
    • Moran model in population genetics
    • PageRank algorithm
    • Markov chain Monte Carlo (MCMC) methods
    • Poisson processes
    • Queuing systems
    • Inspection paradox


  • 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

    • Games and gambling strategies
    • Kolmogorov's zero-one law
    • Actuarial risk modelling
    • Strong law of large numbers
    • Branching processes
    • Statistical hypothesis testing
    • Filtering problems in statistics
    • Secretary problem


Blackboard lectures, in-class questions and discussions



Weitere Informationen

In principle, this VO course should run for 6 hours per week, from the beginning of March to the beginning of May. In any case, the schedule is provisional and will be discussed with the students in the first week (for example, we may change date/time or run it for fewer weekly hours over a longer period, e.g. until the end of May). If the provisional schedule does not work for you, please drop an email to the lecturer before the start of the semester and indicate your preference.

Vortragende Personen


LVA Termine

Mo.09:00 - 12:0004.03.2024 - 27.05.2024EI 1 Petritsch HS Vorlesung
Mi.09:00 - 12:0006.03.2024 - 29.05.2024Seminarraum 107/1 Vorlesung
Theorie stochastischer Prozesse - Einzeltermine
Mo.04.03.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.06.03.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.11.03.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.13.03.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.18.03.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.20.03.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.08.04.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.10.04.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.15.04.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.17.04.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.22.04.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.24.04.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.29.04.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mo.06.05.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.08.05.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.13.05.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.15.05.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mi.22.05.202409:00 - 12:00Seminarraum 107/1 Vorlesung
Mo.27.05.202409:00 - 12:00EI 1 Petritsch HS Vorlesung
Mi.29.05.202409:00 - 12:00Seminarraum 107/1 Vorlesung


Written exam. Additional points may be awarded for class participation.

The first exam will take place after the end of the lectures, on a date to be agreed with attending students. If you wish to take the exam in a subsequent date, please get in touch directly with the lecturer and another exam will be scheduled.


Von Bis Abmeldung bis
31.01.2024 08:00 29.03.2024 19:00 29.07.2024 20:00


066 394 Technische Mathematik Gebundenes Wahlfach
066 395 Statistik-Wirtschaftsmathematik Gebundenes Wahlfach
066 453 Biomedical Engineering Keine Angabe
860 GW Gebundene Wahlfächer - Technische Mathematik Keine Angabe


  • 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