# Faculty of Mathematics

### Stochastics

The field of stochastics deals with statistical applications, stochastic algorithms, the approximation of stochastic partial differential equations and the conceptual characterization of random geometries and universal stochastic processes.

Prof. Belomestny works together with engineers from Bochum on a new project in the scope of the SFB 823 “Statistical Modelling of Nonlinear Dynamic Processes” in the statistical modelling of timely and spectrally highly-resolved audio data in hearing aids. This project is concerned with the simplification of musical signals leading to a perception with less severe distortions and less effort associated with hearing. New methods for users of hearing aids (especially of Cochlea implants) are developed and optimized. This increasingly involves the incorporation of a priori knowledge about musical structures and the constraints presented by the hearing aid.

Denis Belomestny and Mikhail Urusov are part of the team that applied for the research training group “Analysing the Interplay of Energy and Finance Markets”. The complexity of the interplay of finance and energy markets demands innovative and interdisciplinary research instruments that can only be provided by an interdisciplinary research group. With their long history, Mathematics and the Faculty of Economics and Business Administration of the University of Duisburg-Essen accommodate a unique combination of research groups for tackling such a research agenda. As invited speaker, Mikhail Urusov held a mini course “A Functional Limit Theorem and Numerical Approximation for Irregular SDEs” at the international summer school on stochastics and financial mathematics 2015 in Sochi.

Optimal stopping under uncertainty is an important topic in financial mathematics. Nowadays, stopping problems arise predominantly in connection to the rating of new finance products. If American options depend on multiple stocks, multi-dimensional stopping problems arise. Finding effective algorithms for this purpose is a challenge, especially if the occurrence of future market scenarios cannot be foreseen with confidence. A novel algorithm for optimal stopping under uncertainties was developed. The common Monte Carlo method for the computation of prizes for finance instruments only functions properly under certain assumptions about the underlying model. In particular, the coefficients of the stochastic differential equation must be regular and are not permitted to grow too quickly. A new algorithm that also works for irregular and fast-growing coefficients was developed.

Concentrated expertise is available in the analysis and numerical analysis of stochastic (partial) differential equations. One of the central research emphases of Prof. Martin Hutzenthaler is the construction of efficiently implementable solutions. Of interest is the question of whether high-dimensional (e.g. d=100) nonlinear, parabolic differential equations can be approximated with a positive polynomial rate. The motivation for this question arises, for instance, in the finance industry where many issues lead to problems of optimal stochastic control, whose solutions are given by nonlinear Hamilton-Jacobi- Bellman PDEs. Another example from physics (e.g. for the flow of fluids and gases) is the search for implementable approximations of semi-linear stochastic PDEs such as the stochastic 2D Navier-Stokes equations converging with positive L2 rate.

Martin Hutzenthaler has been granted a mini workshop on this topic in February 2017 at the Mathematical Research Institute Oberwolfach together with the co-organizers Annik Lang (Chalmers), Lukas Szpruch (Edinburgh) and Larissa Yaroslavtseva (Passau). Martin Hutzenthaler and Anita Winter are active in the Research Training Group RTG 2131 “High-Dimensional Phenomena in Probability – Fluctuations and Discontinuity”, which began in October 2015. The cooperation between different research groups at the TU Dortmund, the University of Bochum in the Ruhr region and the UDE currently offer a broad scientific education and excellent conditions for 11 doctoral students and 2 postdocs via tandem lectures, summer schools, financial support for research stays and guest researchers, allowing them to follow current research trends beyond their own topic. The projects in Essen focus on questions about the investigation of convergence rates. In this context, we concentrate on a paradigm change already visible in the current research directions. Of central importance is, for instance, the investigation of invariance principles in the dependence of geometric identifying quantities of underlying metric measure spaces, and the approximations of stochastic partial differential equations. A combining bracket is given by the Malliavin calculus and Stein’s method.

A further joint research emphasis of Martin Hutzenthaler and Anita Winter lies in mathematical biology. They both participate in the DFG priority programme SPP 1590 “Probabilistic Structures and Evolution” with a total of three projects. In this context, Martin is primarily interested in the question of whether an altruistic gene can succeed in a spatially-structured population in a ‘short’ time. In this context, ‘short’ means a time period which grows at most logarithmically with the population size on an evolutionary timescale. This is motivated by the aim to resolve the apparent contradiction between Darwin’s evolution theory and the existence of altruism. In the literature, it is shown with simplified spin models (among others in some Nature articles) that altruism can succeed. So far it remains unclear, however, if this can also take place in realistic time periods. Anita Winter is interested in modelling phylogenies of RNA viruses which, due to high mutation and replication rates, evolve so fast that evolution and epidemiology takes place on the same timescale. The pathogen pattern – and particularly the topology of the phylogenies – is influenced by the strength of the selective pressure enforced by the corresponding levels of cross-immunity.

The goal of this project is the development of an agent-based parametric model which delivers the different known shapes of virus phylogenies depending on the form of the selection function. In this context, Anita Winter organized the two workshops on “Genealogies of population models with competition” and “Probabilistic models in evolutionary biology” in cooperation with Anja Sturm (University Göttingen), which took place in June 2014 at the University of Duisburg-Essen and in November 2016 at the University Göttingen, respectively.