Diyora Salimova: Deep neural networks in numerical approximation of high-dimensional PDEs
In recent years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a numerous of computational problems including, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations. Such numerical simulations indicate that DNNs seem to admit the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate in such computational problems. In this talk I present our recent result which rigorously proves that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients.
Sara Svaluto-Ferro: Infinite dimensional polynomial jump-diffusions
Abstract: We introduce polynomial jump-diffusions taking values in an arbitrary Banach space via their infinitesimal generator. We obtain two representations of the (conditional) moments in terms of solution of systems of ODEs. These representations generalize the well-known moment formulas for finite dimensional polynomial jump-diffusions. We illustrate the practical relevance of these formulas by several applications. In particular, we consider (potentially rough) forward variance polynomial models and we illustrate how to use the moment formulas to compute prices of VIX options.