In recent years, a number of results has been proven in the literature for strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. On strong approximation of SDEs with a discontinuous drift coefficient Larisa Yaroslavtseva (University of Passau) We will demonstrate that deep neural networks are surprisingly efficient at resolving discontinuities of potentially very high dimensional functions. We will analyse the extent to which deep neural networks are affected by discontinuous target functions if the discontinuity admits certain regularities. In the associated application areas, it is then necessary to finely resolve the singularities. Finally, in the modelling of physical processes, such as fracture mechanics, singularities naturally appear. In classification problems, decision boundaries are often assumed to possess a certain regularity. For example, in image processing, jumps in intensity value are often associated to the boundaries of physical objects and appear along curves. We also provide conditions under which the confidence bands are asymptotically valid.Īpproximation and estimation of functions with structured singularities by deep neural networksįunctions with structured singularities appear in many application domains. Using a spectral estimator of the Lévy density, we propose a novel implementations of multiplier bootstraps to construct confidence bands on a compact set away from the origin. This report will consider the problem of constructing bootstrap confidence intervals for the Lévy density of the driving Lévy process based on highfrequency observations of a Lévy-driven moving average processes. Spectral Bootstrap confidence bands for Lévy-driven moving average processes Tatiana Orlova (University of Duisburg-Essen) Winter Semester 2021/22 ( back to Contents) Oct 12