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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 -
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
(2020)SAM Research ReportReport -
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Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
(2020)SAM Research ReportRecently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz ...Report -
Solving stochastic differential equations and Kolmogorov equations by means of deep learning
(2018)Research ReportReport -