# Mathematics

## Stochastics

The typical problems of economics and engineering sciences rely on the analysis of large quantities of data with a complex structure. Risk management assessments, for example, require both statistical analysis and an estimate of the parameters of financial time series. Given the high-dimension data structures, risk management and econometric analyses also require means of reducing dimension. Classical techniques such as PCA (Principal Component Analysis) and ICA (Independent Component Analysis) work under very strict assumptions on the underlying model and are therefore only useful to a limited extent. Time series are a special kind of data structure. Often, the interesting variables are only observed indirectly, for instance through linear or non-linear maps. While the data can be explained by models with unknown parameters, determining the model parameters often proves to be an ill-posed problem. This is also characteristic of many problems in financial mathematics, for example estimating and calibrating stock market models. The particular challenge in estimating and calibrating stochastic models in financial mathematics arises because the associated inverse problems are nonlinear. These topics are investigated by the research group of Prof. Denis Belomestny in a subproject of the DFG Collaborative Research Centre 823 “Statistical modelling of nonlinear dynamic processes”, which recently received a positive evaluation. This is a joint research project between mathematicians, statisticians and economists from the Universities of Dortmund, Bochum and Duisburg-Essen. Work on the project has also been conducted in international cooperation with Prof. Nan Chen (Hong Kong), Prof. Valentine Genon-Catalot (Paris), and Dr. Aleksandar Mijatovic (London). Prof. Belomestny’s book, which is to be published in 2015, was written in collaboration with Prof. Markus Reiß (Berlin). Within the cooperation with the research group of Prof. Belomestny, Dr. Mijatovic was granted a Humboldt Research Fellowship for experienced researchers in 2014. Prof. Belomestny’s research group is also working on the numerical analysis of stochastic differential equations (Prof. Belomestny, Jun. Prof. Mikhail Urusov), a rapidly growing area of research at the present time. Further research activities include risk management (Dr. Volker Krätschmer) and diffusion processes and their applications (Jun. Prof. Urusov), areas in which international cooperation with London (Prof. Mihail Zervos, Dr. Mijatovic) and Moscow (Prof. Albert Shiryaev) is also ongoing.

Research activity in Prof. Anita Winter’s group centres on the analysis of complex interacting stochastic systems found in mathematical physics and mathematical biology. One focus of the research is on systems and questions motivated by mathematical biology, especially evolution theory and cell biology. This work involves, for example, observing populations of individuals who are characterised by a (biological) type. Migration takes place within a given geographical structure. The individuals reproduce at rates which are dependent on the locally available essential resources and the current size of the populations competing for those resources. Here the researchers are interested in understanding under which conditions on the model parameters individuals of different phenotypes can also coexist over a long period of time. These are the topics on which the research group is taking part in the DFG Collaborative Research Centre/Transregio 12 “Symmetries and Universalities on Mesoscopic systems” with a subproject entitled “Fluctuations and large deviations in nonequilibrium stochastic dynamics”. The joint research project is being conducted with mathematicians and physicists from the Universities of Bochum, Duisburg-Essen, Cologne and the LMU Munich. Because many microorganisms, in particular RNA viruses, evolve so rapidly, evolution and epidemiology take place on the same time scale. The high mutation and replication rates lead to diversity, which impedes the control of epidemics. The pathogen-associated patterns – and in particular the topology of phylogenies – are influenced by the selective pressure exerted by the corresponding level of cross-immunity. Cross-immunity describes the reaction of the host immune system that fights the virus strain and similar variants. Related questions are explored in a subproject, “Modelling of evolving phylogenies in the context of phylogenetic pattern”, of the DFG Priority Programme SPP 1590 “Probabilistic Structures and Evolution”. The research is conducted in close international cooperation (Canada, France, India, Israel and Singapore).

Prof. Martin Hutzenthaler’s research group studies the regularity and efficient approximation of stochastic (partial) differential equations. Because it is not usually possible to approximate expectation values of solutions at an optimal rate using the Euler method, research focuses here on finding more efficient alternatives. Essential to this work is an understanding of the temporal and spatial regularity of the solutions of stochastic differential equations. One important example is the Heston model, which is approximated on a daily basis in the financial sector to set the price of options. The group also researches selective pressure on spatially structured populations. The “Evolution of altruistic defense traits in structured populations” subproject within DFG Priority Programme SPP 1590 “Probabilistic Structures and Evolution” investigates models for genetic material, the carriers of which gain a selective advantage, in spite of a reduced reproduction rate, as a result of a behaviour that is beneficial to the surrounding population.