Department of Mathematics

Courses for Master's students (M.Sc.) offered in Summer Semester 2020

Due to the current situation in connection with the spread of the COVID-19/Corona virus, it will not be possible to start the courses in the form of classroom teaching. On this page we present the courses for graduate students offered by the Department of Mathematics with the essential information (especially: links to the courses in the online teaching platform OLAT).

 

We will use the week from 14.04. to 17.04.2020 to provide our students with the necessary information about the digital courses, to give an introduction to the necessary electronic tools and, if necessary, to provide materials for acquiring missing previous knowledge. The aim should be that all students are ready to start on 20.04.2020, in order to then continue with the full range of courses in digital form (almost certainly).

Mathematikvorlesung an der TUK: Bild aus Hörsaal

Further courses in SS 2020

Information on the courses for undergraduate students (in German language)

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For a better planning of the digital course offerings, we ask you to register by April 8, 2020, 23:00 in URM (https://urm.mathematik.uni-kl.de) for all lectures and seminars you wish to attend in the summer semester 2020!

Financial Mathematics

Contents

  • Basics of stochastic analysis (Brownian motion, Itô-integral, Itô-formula, martingale representation theorem, Girsanov theorem, linear stochastic differential equations, Feynman-Kac formula)
  • Diffusion model for share prices and trading strategies
  • Completeness of market
  • Valuation of options with the replication principle, Black-Scholes formula
  • Valuation of options and partial differential equations
  • Exotic options
  • Arbitrage bounds (Put call parity, parity of prices for European and American calls)

Contact time

4 SWS lecture
2 SWS tutorials

The lecture is offered every year in the summer term.

Prerequisites with regard to contents

Course "Probability Theory"

Links

Here you find the KIS entry: Financial Mathematics (lecture) Financial Mathematics (tutorial)

Here you find the OLAT course: TUK Financial Mathematics SS2020

Functional Analysis

Content

  • Hahn-Banach theorem and its applications
  • Baire category theorem and its applications (uniform boundedness principle, Banach-Steinhaus theorem, open mapping theorem, inverse mapping theorem, closed graph theorem)
  • weak convergence (Banach-Alaoglu theorem, reflexive Banach spaces, lemma of Mazur and its applications)
  • projections (closed complement theorem)
  • bounded operators (adjoint operators, spectrum, resolvent, normal operators)
  • compact operators (Fredholm operators, Fredholm alternative and its applications, spectral theorem (Riesz-Schauder) and applications to normal operators)

contact time

4 SWS lecture
2 SWS exercise classes

substantive prerequisites

content of the introductory lecture "Einführung in die Funktionalanalysis" as well as concepts from "Maß- und Integrationstheorie"

Life Insurance Mathematics

Contents

  • Elementary financial mathematics (calculation of interest)
  • Mortality
  • Insurance benefits
  • Net premiums and net actuarial reserves
  • Inclusion of costs
  • Life related insurance
  • Various reject causes

Contact time

2 SWS lecture

The lecture is offered every year in the summer term. It takes place during the second half of the semester.

Prerequisites with regard to contents

Course "Stochastic Methods" from the Bachelor's degree programme.

Links

Here you find the KIS entry: Life Insurance Mathematics (lecture)

Here you find the OLAT course: TUK Life Insurance Mathematics SS2020

Markov Switching Models and their Applications in Finance

Contents

  • Discrete-time and continuous-time Markov chains
  • Hidden Markov models in discrete time
  • Continuous-time Markov switching models
  • Parameter estimation and filtering
  • Modelling financial asset prices
  • Econometric properties of financial time series and model extensions
  • Applications to portfolio optimization

Contact time

2 SWS lecture

The lecture is offered on an irregular basis.

Prerequisites with regard to contents

Module "Mathematical Statistics" or "Probability Theory"

Monte Carlo Algorithms

Content

Monte Carlo algorithms are the algorithms which use randomness. The course gives an introduction to this important basic algorithmic technique in mathematics and computer science.

It discusses the topics

  • Direct Simulation
  • Simulation of distributions
  • Variance reduction
  • Markov Chain Monte Carlo algorithms
  • High-dimensional integration

and applications in physics as well as in financial and actuarial mathematics

Contact Time

4 SWS / 60 h Lectures
2 SWS / 30 h Exercise Classes

Prerequisites

course „Stochastische Methoden“ and basic knowledge og in numerical methods

Frequency

The lecture is offered irregularly (in summer semester)

Operator Semigroups and Applications to PDE

Inhalte

  • Definitionen, Generatoren, Resolventen, Beispiele, •
  • Hille-Yosida Theorem, Lumer-Phillips Theorem, •
  • Kontraktions-Halbgruppen, Analytische Halbgruppen, Operator-Gruppen, •
  • Approximationen, Störungen, •
  • Anwendungen auf Partielle Differentialgleichungen (u.a. Wärmeleitungsgleichungen, Wellengleichungen, Schrödinger-Gleichungen).

Kontaktzeit

4 SWS Vorlesung
2 SWS Übung

Inhaltliche Voraussetzungen

Lehrveranstaltung "Functional Analysis"

Theory of Scheduling Problems

Content

  • Classification of scheduling problems,
  • The link between scheduling and combinatorial optimization problems,
  • Single machine problems,
  • Parallel machines,
  • Job shop scheduling,
  • Due-date scheduling,
  • Time-Cost tradeoff Problems.

Contact Time

4 SWS / 60 h Lectures
2 SWS / 30 h Exercise Classes

Prerequisites (Content)

Basic lectures in analysis and linear algebra, knowledge in Optimisation and Stochastics (e.g. from courses „Lineare und Netzwerkoptimierung“ and „Stochastische Methoden“)

Frequency

each summer semester

Links/Contact

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