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Uniform error estimates for artificial neural network approximations for heat equations
(2019)SAM Research ReportRecently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature which examine the approximation capabilities of such deep learning based approximation ...Report -
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On existence and uniqueness properties for solutions of stochastic fixed point equations
(2019)SAM Research ReportThe Feynman--Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence ...Report -
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
(2019)SAM Research ReportReport -
Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise
(2019)SAM Research ReportReport -
Space-time error estimates for deep neural network approximations for differential equations
(2019)SAM Research ReportReport -
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Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
(2019)SAM Research ReportReport -