Mathematics

Many problems occurring in economics and life sciences, in particular modelling and monitoring of financial markets and risk assessment in insurance and the financial system, require analysis of large data sets with a complex structure. Risk management analysis, for instance, relies on statistical analysis and estimating the parameters of financial time series as well as the use of Monte Carlo methods combined with regression methods. Risk management and econometric analysis furthermore require methods of dimension reduction on account of the high dimensionality of the data structures. Classical tools such as PCA (Principal Component Analysis) and ICA (Independent Component Analysis) are based on very strict model assumptions and are therefore only of limited use. Time series are a special type of data structure. Often, the interesting quantities are only observed indirectly, for instance through linear or non-linear maps. The data can be explained by models with unknown parameters, but determining the model parameters is often an ill-posed problem. This is also characteristic of many problems in financial mathematics, for example estimation and calibration of stock market models. The problem of calibrating ­stochastic models in financial mathematics is a particular challenge, because the inverse problems involved are non-linear. These topics are investigated by the research group of Professor Denis Belomestny in a subproject of DFG Collaborative Research Centre SFB 832 “Statistical modelling of nonlinear dynamic processes”. This is a joint ­research project between mathematicians, ­statisticians and economists from the Universities of Dortmund, Bochum and Duisburg-Essen.
The activities of the research group of Professor Anita Winter focus on the analysis of complex interacting stochastic systems which arise in mathematical physics or mathematical biology. Of particular importance are those systems and questions that are motivated by mathematical ­biology, especially evolution theory and cell biology. One example considers populations of individuals which are characterized by a (biological) type. Within a predefined geographic structure migration occurs. The individuals reproduce at rates dependent on the locally available essential resources and on the current size of the population competing for those resources. The researchers are interested in understanding under which conditions on the parameters of the model it is possible for individuals of different phenotypes to coexist even over a long period of time. Such problems are the topic of the “Fluctuations and large deviations in nonequilibrium stochastic dynamics” subproject within DFG Collaborative Research Centre/Transregio SFB/TR 12 “Symmetries and Universalities in Mesoscopic systems”. This is a joint research project between mathematicians and physicists from the Universities of Bochum, Duisburg-Essen, Cologne and the LMU Munich.
Many microorganisms, in particular RNA ­viruses, evolve so fast that evolution and epidemiology take place on the same time scale. The high rates of mutation and replication lead to diversity, which impedes the control of epidemics. The pathogen-associated patterns – and in ­particular the topology of the phylogenies – are influenced by the selective pressure exerted by the corresponding level of cross-immunity. Here, cross-immunity refers to the reaction of the immune system of the host which fights the virus strain and similar variants. Related questions are analyzed by the research group in the “Modelling of evolving phylogenies in the context of phylogenetic pattern” subproject of DFG Priority Programme 1590 “Probabilistic Structures and Evolution”. This research is conducted in close international cooperation with groups in Canada, France, India, Israel and Singapore.