Dependence Modelling by Lévy Processes with Applications to Mathematical Finance, Insurance and Econometrics

Claudia Klüppelberg
Technische Universität München
Claudia Kluppelberg

Syllabus

Stochastic models driven by a general Lévy process play an increasingly important role in many applications areas, prominently also in the mathematics and statistics for finance and insurance. In contrast to the classical Brownian motion driven models they allow for jumps - a phenomenon obvious in insurance and operational risk data, but also in volatilities and price processes on financial markets. Special tests have been developed to undermine this fact. This has revolutionized the core of mathematical finance in the sense that pricing and hedging formulas nave no closed forms as in the Black-Scholes model, but simulation and numerical methods had to be developed to obtain more realistic prices. This development has also revealed the fact that risks cannot be hedged completely.

The development on the econometric side has been triggered by the fact that data from liquid markets are high-frequently sampled at not necessarily equidistant intervals. The development of continuous-time models, which fit the stylized facts of high-frequency data has been met with a zoo of new models and methods for their statistical fitting. The new deregulated electricity markets present new interesting challenges. Of course, Lévy processes are at the core of this development.

Lévy processes are Markov processes, which are nice models, when it comes to statistical estimation and price calculations, but they may not in all situations be the most realistic ones. It has been observed for a long time that volatility exhibits some long range dependence effect. Moreover, macroeconomic variables often serve as latent processes, when for instance credit risk models are suggested. There is strong statistical evidence that processes like supply and demand, interest rates and the gross national product exhibit some long range dependence effect, which has to be taken into account. Fractional versions of Lévy processes offer a welcome extension away from fractional Brownian motion, similarly as Lévy processes extend Brownian motion.

In this course we will present the most important dynamic models for insurance and finance. Besides providing the necessary mathematical background we will also focus in particular on modelling issues and the interpretation of different features of the models under consideration.

Time and Place

Course meets Monday (2009/11/16 and 2009/11/23) from 10.15 to 11.45 am and Wednesday (2009/11/18 and 2009/11/25 and 2009/12/2), from 8.30 to 10.00 am, Gr. Saal, Endenicher Allee 60. (These lectures are open to the interested public and there is no fee for attending.)
Course Material
Relevant papers can be downloaded from the publication page of Claudia Klüppelberg.

A joint activity of the Hausdorff Center for Mathematics and the Bonn Graduate School of Economics.