Detailed information can be found in the course descriptions and in the module handbooks (in German only).
Lecture: Mo, Mi, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Teacher: Ernst August v. Hammerstein
Assistant: Sebastian Stroppel
Language: in English
The lecture builds on basic knowledge about Probability Theory. The fundamental problem of statistics is to infer from a sample of observations as precise as possible statements about the data-generating process or the underlying distributions of the data. For this purpose, the most important methods from statistical decision theory such as test and estimation methods are introduced in the lecture. \\ Key words hereto include Bayes estimators and tests, Neyman-Pearson test theory, maximum likelihood estimators, UMVU estimators, exponential families, linear models. Other topics include ordering principles for reducing the complexity of models (sufficiency and invariance). Statistical methods and procedures are used not only in the natural sciences and medicine, but in almost all areas in which data is collected and analyzed This includes, for example, economics (“econometrics”) and the social sciences (especially psychology). However, in the context of this lecture, we will focus less on applications, but---as the name suggests---more on the mathematical justification of the methods.
Required: Probability Theory (in particular measure theory and conditional probabilities/expectations)
Advanced Lecture in Stochastics
Elective in Data
Questions sesssion / flipped classroom: Mo, 10-12h, HS II, Albertstr. 23b
Letcure (4 hours): asynchronous videos
Teacher: Peter Pfaffelhuber
Assistant: Samuel Adeosun
Language: in English
A stochastic process \((X_t)_{t\in I}\) is nothing more than a family of random variables, where \(I\) is some index set modeling time. Simple examples are random walks, Markov chains, Brownian motion and derived processes. The latter play a particularly important role in the modeling of financial mathematics or questions from the sciences. We will first deal with martingales, which describe fair games. After constructing the Poisson process and Brownian motion, we will focus on properties of Brownian motion. Infinitesimal characteristics of a Markov process are described by generators, which allows a connection to the theory of partial differential equations. Finally, a generalization of the law of large numbers is discussed with the ergodic theorem for stationary stochastic processes. Furthermore, insights are given into a few areas of application, such as biomathematics or random graphs.
Required: Probability Theory I
Advanced Lecture in Stochastics
Elective in Data
Lecture: Mo, Mi, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Thorsten Schmidt
Assistant: Moritz Ritter
Language: in English
This lecture marks the culmination of our series on probability theory, achieving the ultimate goal of this series: the combination of stochastic analysis and financial mathematics---a field that has yielded an amazing wealth of fascinating results since the 1990s. The core is certainly the application of semimartingale theory to financial markets culminating in the fundamental theorem of asset pricing. This results is used everywhere in financial markets for arbitrage-free pricing.
After this we look into modern forms of stochastic analysis covering neural SDEs, signature methods, uncertainty and term structure models. The lecture will conclude with an examination of the latest applications of machine learning in financial markets and the reciprocal influence of stochastic analysis on machine learning.
Required: Probability Theory II (Stochastic Processes)
Advanced Lecture in Stochastics
Elective in Data