This course provides an introduction to the theory of stochastic processes. Topics include:

  • Conditioning on sigma-fields
  • Stochastic processes: basic notions and examples
  • Construction of stochastic processes via the Kolmogorov extension theorem
  • Martingales
  • Martingal convergence theorems and applications
  • Optional sampling theorems
  • Brownian motion
  • Regularity of Brownian paths and the law of the iterated logarithm
  • Skorokhod embedding
  • Donsker's invariance principle
  • (Strong) Markov property
  • Applications of Brownian motion to partial differential equations

Prerequisites are Analysis I-III, Linear Algebra I-II, Stochastics I (or equivalent courses).

The aim of this course is to provide an introduction to continuous-time Markov processes. The included topics are:

  • Definition and existence of Markov processes
  • Strong Markov property
  • Lévy processes as examples of (strong) Markov processes
  • Contraction semigroup and Infinitesimal generator
  • Feller semigroup, Feller processes and the existence of their cadlag modifications.
  • Martingale problems

The prerequisite course is "Stochastics I". 

The course "Stochastics II" is helpful but not obliged.

The course provides an introduction to Ito's stochastic calculus and stochastic differential equations driven by a Brownian motion. Topics include:

  • Linear differential equations with additive noise
  • Integration with respect to a Brownian motion (Ito calculus)
  • Strong solutions of stochastic differential equations
  • Weak solutions of stochastic differential equations
  • Applications to option pricing

Course prerequisites are "Stochastics I" and "Stochastics II".

Die Vorlesung führt in die maßtheoretische Wahrscheinlichkeitstheorie ein. U.a. werden die folgenden Themen behandelt:

  • Allgemeine Wahrscheinlichkeitsräume
  • Zufallsvariablen und deren Verteilung
  • Bedingen auf Ereignisse
  • Unabhängigkeit
  • Erwartungswert, Varianz und Kovarianz
  • Summen von unabhängigen Zufallsvariablen
  • Charakteristische Funktionen
  • Konvergenzbegriffe für Folgen von Zufallsvariablen und Folgen von Wahrscheinlichkeitsmaßen
  • Gesetze der großen Zahlen
  • Zentraler Grenzwertsatz

Die Vorlesung setzt Grundkenntnisse aus der Analysis (Analysis I, II) und Linearen Algebra (Lineare Algebra I) voraus. Obwohl wichtige Elemente der Maß- und Integrationstheorie kurz wiederholt werden, gehört das Modul "Analysis III" zu den empfohlenen Vorkenntnissen.