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Exponential moments for numerical approximations of stochastic partial differential equations
(2016)Research reports / Seminar for Applied MathematicsReport -
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Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
(2014)SAM Research ReportStrong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates ...Report -
Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
(2018)SAM Research ReportReport -
Generalised multilevel Picard approximations
(2019)SAM Research ReportIt is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific literature suffer under the curse of dimensionality in the sense that the number of computational operations needed to compute an approximation with an error of size at most $ \epsilon > 0$ grows at least ...Report -
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 -
Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
(2015)SAM Research ReportStrong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for ...Report -
Solving high-dimensional optimal stopping problems using deep learning
(2019)SAM Research ReportNowadays many financial derivatives which are traded on stock and futures exchanges, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlyings in the associated hedging portfolio. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due ...Report