Mathematical study programmes: courses taught in English 1. Master 1.1.Winter term An Introduction to Measure-Theoretic Probability o ECTS: 4 o Hours per week: lecture (2 hours) and exercise (1 hour) o Contents Definition and properties of measure and Lebesgue integral The fundamentals of probability: probability space, random variables, conditional expectation, modes of convergence, convolutions and characteristic functions, central limit theorem The fundamentals of statistics: simple random sampling, introduction to estimation techniques o Prerequisites: Analysis I and II Linear Algebra I and II Elementary probability theory and statistics Asset Pricing o Lecturer: Prof. Günther Löffler o Hours per week: lecture (4 hours) and exercise (1 hour) o ECTS: 7 Time value of money, compounding, measuring return, discount rates, dividend discount model, expected utility The stochastic discount factor Using the stochastic discount factor approach to understand returns on risky and risk-free assets Factor pricing models The CAPM and the empirical evidence, Fama-French-3-factors Aggregate stock price behavior Equity premium puzzle, time series predictability Rationality and Behavioral Finance Bubbles, Prospect Theory Insurance economics Topics in Financial Mathematics o Hours per week: lecture (4 hours) and exercise (2 hours), tutorial (2 hours) Several different courses are offered in an irregular sequence. The module can be attended several times (in different courses). Possible courses include: Diffusion Processes with Applications to Finance 1
Time Series Analysis Non-linear Time Series Analysis Levy Processes, Stochastic Analysis and Financial Mathematics Numerical Finance Asymptotic Methods in Insurance Interest Rate Modeling Stochastic Risk Theory Risk Management in Insurance o Lecturer: Prof. An Chen o ECTS: 7 o Time: lecture (4 hours), exercise (2 hours) o Content: This course provides an introduction to the basics of risk management in insurance. Topics include: Interest rate risk (yield curves, forward rates, caps and floors) Occupational pension plans (defined benefit, defined contributions) State preference theory (state prices, arbitrage, market completeness, Arrow-Debreu and complex securities) Equity options (Introduction to option theory, Black-Scholes-Merton model) Portfolio planning (Markowitz portfolio theory, CAPM, optimal investment decision, optimal consumption-savings decision) Economic Capital, RAROC o Prerequisites: Lectures in insurance economics and insurance finance are useful 1.2.Summer term Financial Mathematics II o Time: lecture (4 hours), exercise (2 hours) o Hours per week: lecture (4 hours) + exercise (2 hours) Stochastic analysis: Stochastic integration, stochastic differential equations,(semi-)martingales Continuous-time financial market models: Valuation and hedging of derivatives in complete and incomplete financial markets, stochastic volatility Interest rate models: Term structure modeling, interest rate derivatives, LIBOR Market models 2
Risk Theory o Lecturer: Prof. Evgeny Spodarev o Hours per week: lecture (4 hours) +exercise (2 hours) This course provides an introduction to the mathematical models of non-life insurance with emphasis on Distribution of claim sizes Distribution of the number of claims Distribution of the aggregate claim amount Simulation Premium calculation Reinsurance Ruin probabilities Practical Financial Engineering o ECTS: 5 o Hours per week: lecture (2 hours) and exercise (1 hour) o Contents Use of a financial information system to obtain prices of standard or complex financial assets; Pricing and hedging of standard or complex derivative instrumentsapplication of standard or advanced techniques; Advanced stochastic simulation/numerical routines in Financial mathematics o Prerequisite: Financial Mathematics I Risk Management Roundup (seminar) o ECTS: 5 o Hours per week: 2 o Contents Mathematical techniques in risk assessment and management. Key concepts used in modern risk management. Factors determining successful risk Management Organizational aspects and limitations of risk models o Prerequisites: Financial Mathematics I and Asset Pricing Survival and Event History Analysis o Lecturer: Prof. Jan Beyersmann o ECTS:9 o Hours per week: lecture (4 hours), exercise (2 hours) o Every other winter term Time-to-event data are omnipresent in fields such as medicine, biology, 3
o demography, sociology, economics and reliability theory. In biomedical research, the analysis of time-to-death (hence the name survival analysis) or time to some composite endpoint such as progression-free survival is the most prominent advanced statistical technique. At the heart of the statistical methodology are counting processes, martingales and stochastic integrals. This methodology allows for the analysis of time-to-event data which are more complex than composite endpoints and will be the topic of this course. The relevance of these methods is, e.g. illustrated in the current debate on how to analyse adverse events. Time permitting, we will also discuss connections between causal modelling and event histories. Prerequisites: Basic knowledge of standard survival analysis and of R is helpful. Statistical Methods of Risk Theory o Lecturer: Prof. Evgeny Spodarev o Hours per week: lecture (4 hours), exercise (2 hours) o Every other winter term 1. Bayes estimators and credibility 1.1 Bayes estimators 1.2 Exponential dispersion families 1.3 Conjugate priors 1.4 Conditional expectation 1.5 The Bühlmann-Straub-estimator 2. Hypotheses tests 3. Multivariate linear models, generalized linear models 4. Biometric actuarial bases 5. Data mining (Principal component analysis) o Prerequisites Basic knowledge of probability calculus and statistics as taught, for example, in "Elementare Wahrscheinlichkeitsrechnung und Statistik" and "Stochastik I". "Risk Theory I" is helpful but not necessary Multivariate Analysis o Lecturer: Prof. Markus Pauly o Hours per week: lecture (4 hours, exercise: 2 hours, tutorial 1 hour) Data visualization. How to present multivariate data? Hypothesis construction and testing; e.g. likelihood-ratio-tests and nonparametric tests Confidence ellipsoids Dimension reduction or structural simplification and their limitations Investigation of dependence among variables Bootstrap for multivariate data Classification and prediction 4
o Prerequisites: Analysis I-II; Linear Algebra I-II; Stochastics I Elementary Probability and Statistics (or courses of similar content) 1.3. Winter and summer term Topics in Life and Pension Insurance o Lecturer: Prof. An Chen o ECTS: 7 o Hours per week: lecture (2 hours) exercise (1 hour) o Prerequisites: knowledge in insurance mathematics General issues concerning life and pension insurance Models for life annuities and life insurance Black-Scholes model Risk-neutral pricing Default risk modeling Optimal asset allocation 2. Bachelor and Master 2.1.Winter term Stochastics II o Lecturer: Prof. Evgeny Spodarev o Hours per week: lecture (4hours) and exercise (2hours) Counting processes and renewal processes; Poisson point processes Wiener process Martingales Lévy processes Stationary processes in discrete time Stochastics III o Lecturer: Prof. Markus Pauly o ECTS: 4 o Hours per week: lecture (2 hours) and exercise (2 hours) Multivariate normal distribution Linear und general linear models Tests for distributional assumptions Non-parametric localization tests o Prerequisites: probability calculus and Stochastics I 5
Optimization II o Lecturer: Prof. Dieter Rautenbach o Hours per week: lecture (4 hours) and exercise (2 hours) P vs NP Branch & bound Approximation algorithms Semidefinite programming o Prerequisite: Optimization I Financial Mathematics I o Lecturer: Prof. Alexander Lindner o Hours per week: lecture (4 hours) + exercise (2 hours) Financial market models in discrete time: arbitrage freeness and completeness Valuation of European, American and path-dependent options Foundations of continuous time market models and of the Black- Scholes model Interest rate models and derivatives Risk measures Portfolio optimisation and CAPM o Prerequisites Analysis 1 and 2 Linear Algebra Stochastics1 Elementary probability theory and statistics Measure theory 2.2.Summer term Insurance Economics o ECTS: 6 o Lecturer: Prof. An Chen o Hours per week: lecture (2 hours), exercise (1 hour) This course provides an introduction to the basics of insurance economics. The topics addressed include: Basics of insurance economics Choice under uncertainty (Expected utility theory and rational decision under risk, measure for risk aversion, mean variance preferences) Insurance demands by households (Base model, insurance demand without fair premium, pareto-optimal insurance contract) 6
Insurance demand by firms (Risk management and diversification, Risk management forward, future and options, Corporate demand for insurance) Insurance supply (Traditional premium calculation, financial modelling of insurance pricing, economies of scope, economies of scale) Microeconomic analysis (moral hazard, adverse selection) Insurance regulation Introduction to Meta-Analysis o ECTS: 4 o Lecturer: Prof. Jan Beyersmann o Hours per week: lecture (2 hours), exercise (1 hour) o Content: Meta-analysis is the statistical methodology for combining quantitative evidence from studies. Meta-analysis is very prominent in medical research, where it features in almost every systematic review, but it is also attracting increasing interest in other fields such as economics. Major practical concerns which fuel methodological research to this day are the need to detect and adjust for possible biases such as publication bias, synthesis from studies which compared different combinations of interventions or studies that measured interventions on different scales or with different outcomes. 3. Bachelor Winter term Introduction to Survival Analysis o ECTS: 5 o Hours per week: lecture (2 hours) and exercise (1 hour) o Lecturer: Prof. Jan Beyersmann o Content Time-to-event data are ubiquitous in fields such as medicine, biology, demography, sociology, economics and reliability theory. In biomedical research, the analysis of time-to-death (hence the name survival analysis) or time to some composite endpoint such as progression-free survival is the most prominent advanced statistical technique. One distinguishing feature is that the data are typically incompletely observed - one has to wait for an event to happen. If the event has not happened by the end of the observation period, the observation is said to be right-censored. This is one reason why the analysis of time-to-event data is based on hazards. This course will emphasize the modern process point of view towards survival data without diving too far into the technicalities. o Prerequisites The level of the course is that of a last year's bachelor course in Mathematical Biometry. Some basic programming knowledge in R would be helpful. 7
Graph Theory o Lecturer: Prof. Dieter Rautenbach o Hours per week: lecture (4 hours), exercise (2 hours) o Content The lecture 'Graph Theory' is one of the basic lectures at our institute. It is necessary for the lecture 'Graph Theory 2' and helps to understand the contents of the lectures 'Optimization I', 'Optimization II', 'Probabilistische Methoden' as well as the whole area of discrete mathematics. Many problems in graph theory are easily explained to non-experts - even to nonmathematicians. For example, consider the problem of coloring a map where every country receives one color but two countries with a common border receive distinct colors. How many colors do we need at most? It took almost 100 years until Appel and Haken solved this problem by proving that four colors always suffice. Their proof is extremely long and sophisticated. Nevertheless, as one highlight of the lecture we will prove a weakening of the 4-color-theorem, namely that five colors suffices. Knowing the basic concepts of graph theory allows to describe many real world problems. In addition, there are many good possible topics in graph theory for bachelor and master theses. 8