Information on the lecture Lévy Processes and Financial Applications (SS 2025)

Lecturer: Dr. Ernst August v. Hammerstein

Assistant: Sebastian Hahn, M.Sc.

Date: Mondays, 14-16h, HS II, Albertstr. 23b

Exercise: Wednesdays, 16-18h, SR 403, Ernst-Zermelo-Str. 1

ECTS: 6 points

Language: english

Lévy processes are the continuous-time analogues of random walks in discrete time as they possess, by definition, independent and stationary increments.
They form a fundamental class of stochastic processes which has widespread applications in financial and insurance mathematics, queuing theory, physics and telecommunication. The Brownian motion and the Poisson process, which may already be known from other lectures, also belong to this class. Despite their richness and flexibility, Lévy processes are usually analytically and numerically very tractable because their distributions are generated by a single univariate distribution which has the property of infinite divisibility.

The lecture starts with an introduction into infinitely divisible distributions and the derivation of the famous Lévy-Khintchine formula. Then it will be explained how the Lévy processes emerge from these distributions and how the characteristics of the latter influence the path properties of the corresponding processes. Finally, after a short look at the method of subordination, option pricing in Lévy-driven financial models will be discussed.

News

The ILIAS page for this course is linked to HISinOne. When you register for this lecture in HISinOne, you will automatically be registered in the corresponding ILIAS course, an access password for the latter is not required!

Tutorial and exercise sheets

The lecture will be accompanied by a weekly tutorial in which solutions to the exercise sheets will be discussed and further questions concerning specific lecture contents can be answered. Currently, the tutorial is scheduled for
Wednesdays from 16-18h in SR 403, Ernst-Zermelo-Str. 1

The weekly exercise sheets will be uploaded on ILIAS on Mondays after the lecture. You should submit your solutions until the same day in the subsequent week by uploading them in digital form on ILIAS.

Examination (Prüfungsleistung) and pass/fail assessment (Studienleistung)

To obtain the ECTS points for this course, you have to succeed in the pass/fail assessment and---depending on your course of studies and the module you choose for this course---also pass a final oral examination (duration approx. 30 minutes).
For the pass/fail assessment, you should earn at least 50% of the maximally accessible exercise points and additionally present one solution of an exercise in the tutorial.

Necessary prior knowledge

Probability Theory
Knowing some further concepts from Probability Theory II (stopping times, filtrations) may be helpful

Literature

D. Applebaum: Lévy processes and Stochastic Calculus, Cambridge University Press, 2005.
J. Bertoin: Lévy Processes, Cambridge University Press, 2005.
R. Cont, P. Tankov: Financial Modelling with Jump Processes, Chapman & Hall/CRC, 2004.
E. Eberlein, J. Kallsen: Mathematical Finance, Springer, 2019.
P. E. Protter: Stochastic Integration and Differential Equations (Second Edition, Version 2.1), Springer, 2005.
K.-I. Sato: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.

Usability

Wahlmodul im Optionsbereich (2HfB21)
Wahlpflichtmodul Mathematik (BSc21)
Mathematische Ergänzung (MEd18)
Angewandte Mathematik (MSc14)
Mathematik (MSc14)
Vertiefungsmodul (MSc14)
Wahlmodul (MSc14)
Elective in Data (MScData24)

Consultation hour

Lecturer consultation hours: Thursdays 10-11 a.m.
For shorter questions, you can also send an email to ernst.august.hammerstein@stochastik.uni-freiburg.de
Assistent consultation hours: by appointment