Numerical Methods for Stochastic Differential Equations with Irregula...

SDEs with irregular coefficients are currently of high interest. There are some specific types of irregularities leading to different problems when studying convergence of numerical schemes. These types of irregularities are usually studied separately in the literature. Examples are polynomially growing coefficients, or discontinuous coefficients. We consider SDEs that suffer from both of these types of irregularities and study strong convergence of the tamed Euler–Maruyama scheme. This is joint work with Michaela Szölgyenyi (University of Klagenfurt). In the second part of the talk, we focus on deep neural networks (DNNs), which have been successfully used in many computational problems including, for example, fraud detection or pattern recognition. DNN algorithms have been also proven to be enormously successful in overcoming the curse of dimensionality, in particular for solving Kolmogorov-type partial differential equations in hundreds of dimensions in reasonable computation time. Nothing is known until now on using neural networks in connection with the so-called Wong-Zakai method that approximates stochastic differential equations by suitable random ordinary differential equations. We are exploring whether neural networks are numerically beneficial in this context and provide an algorithm for that. This is joint work with Andreas Neuenkirch (University of Mannheim) and Michaela Szölgyenyi (University of Klagenfurt).