Information about the Contents of the Standard Lectures
Prerequisites for the lectures are a course in basic probability theory and basic knowledge in measuretheory. The lectures offer an introduction into the theory of stochastic processes with indepth studyof some fundamental classes of stochastic processes and their applications.
- Definition of stochastic processes, construction and existence, classification.Kolmogorov and Ionescu-Tulcea theorems.
- Markov chains with discrete time and discrete state space.Classification, stopping times and strong Markov property, asymptotic behaviour and limitingdistributions, equilibrium behaviour and stationary distributions.
- Markov processes with continuous time and discrete state space.Classification, analytical properties of the transition semigroup, infinitesimal generator, back-ward and forward differential equation, explosion in finite time, stopping times and strongMarkov property, asymptotic behaviour and limiting distributions, equilibrium behaviour andstationary distributions. Birth-death processes.
- Martingales with discrete time.Definition of martingales, sub- and supermartingales, stopping times, optinal sampling andoptional stopping, limit theorems, representation of uniformly integrable martingales.
- Stochastic processes with stationary and independent increments (Levy processes).General properties, infinitely divisible distributions, Poisson process, Brownian motion.
- General Markov processes and Markovian semigroups.Semigroups associated with a Markov process. Infinitesimal generator and differential equa-tions, convolution semigroups, the problem of continuity and differentiabilty of paths.
- Renewal theory and renewal processes.Counting processes, renewal processes, renewal function, renewal measure, renewal equation,residual lifetime process, renewal paradoxon, general regenerative processes, limit theorems,limiting distributions for age processes.
[Asm87] S. Asmussen. Applied Probability and Queues. John Wiley and Sons, Inc., Chichester –New York – Brisbane – Toronto – Singapore, 1987.
[Bre68] L. Breiman. Probability. Addison–Wesley Publishing Company, Reading, Massachusetts– Menlo Park, California – London – Don Mills, Ontario, 1968.
[Chu67] K. L. Chung Markov Chains with Stationary Transition Probabilities. Springer–Verlag,Berlin – Heidelberg – New York, 1967.
[KT75] S. Karlin and H. M. Taylor. A First Course in Stochastic Processes. Academic Press,New York – San Francisco – London, second edition, 1975.
[KT81] S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press,New York – San Francisco – London, 1981.
[Res92] S. Resnick. Adventures in Stochastic Processes. Birkhäuser, Boston, 1992.
[RSST99] T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels. Stochastic Processes for Insuranceand Finance. Wiley Series in Probability and Statistics. Wiley, Chichester, 1999.
The prerequisite for these lectures is some basic knowledge in measure theoretic probability. It is assumed that notions like σ–algebra, measure, Lebesgue measure, measure integral, independence, weak law of large numbers, central limit theorem, Markov kernel, Fubini's
theorem etc. are known.
The lectures offer basic results on estimation and testing for parametric statistical models. They are accompanied by exercises (Ubungen) which are an indispensable part of the studies.
- Optimal unbiased estimators
- The substitution principle for estimators and Maximum–Likelihood estimators
- The multivariate normal distribution
- The general linear model. (Basic results on testing and estimation without optimality considerations, e.g. the Gauss–Markov theorem for estimators and Fishers F–test)
- Optimal test for simple hypotheses (Neyman–Pearson Lemma)
- Monotone likelihood ratio and optimal one–sided tests
- Optimal two–sided tests
- Conditional expectation and conditional experiments (Radon–Nikodym theorem)
- Sufficiency and completeness and their application for estimating and testing
- Exponential families
- Testing in κ+ 1–parametric exponential families: Conditional tests
- Transformation to unconditional tests
- Testing for the mean and the variance in normal families (Optimality of Student's t–test etc.)
- Duality of confidence regions and families of tests.
- Bickel, P. und Doksum,K.(1977) Mathematical Statistics. Holden–Day, Diisseldorf
- Ferguson,Th. S. (1967) Mathematical Statistics. Academic Press, New York
- Lehmann,E. L. (1959) Testing Statistical Hypotheses. Wiley, New York
- Lehmann, E. L. (1983) Theory of Point Estimation.Wiley, New York
- Pfanzagl, J. (1994) Parametric Statistical Theory. De Gruyter, Berlin
- Witting, H. (1974) Mathematische Statistik.Teubner, Stuttgart
- Witting, H. (1985) Mathematische Statistik I. Teubner Stuttgart
The prerequisites for these lectures is a course in basic probability theory and a basic knowledge in measure theory.
The lectures offer an introduction to actuarial mathematics of life insurances.
- Elementary financial mathematics.
Discussion of different ways to describe interest payments, the valuationof payment flows and the concept of the present value.
- Stochastic models of the risk insured and the insurance contract in thesettings „one person under single risk”, „several persons under singlerisk“ and „one person under competing risks“.
- Basic principles of premium calculation.
The principle of equivalence for stochastic payment flows and the equi–valence premium.
- The dynamics of the reserve process.
Recursions for the reserve process in discrete time and Thiele’s integralequations in continuous time.
- Analysis of the loss process.
Hattendorff’s theorem on the decomposition of the loss process intolosses in different years and states of the policy.
- H. MILBRODT und M. HELBIG (1999). Mathematische Methoden der Personenversicherung. de Gruyter, Berlin.
- M. KOLLER (2000). Stochastische Methoden in der Lebensversicherung. Springer, Berlin.
- H.U. GERBER (1995). Life Insurance Mathematics (2nd. ed.). Springer, Berlin