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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 -
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 -
Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise
(2015)SAM Research ReportStochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the ...Report -
On the differentiability of solutions of stochastic evolution equations with respect to their initial values
(2016)Research reports / Seminar for Applied MathematicsReport -
Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions
(2016)Research ReportWe show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process which is strongly H¨older continuous in time, then this sequence converges in the strong sense even with respect to much stronger H¨older norms and the convergence rate is essentially reduced by the H¨older exponent. Our first application hereof establishes pathwise convergence rates of spectral Galerkin ...Report -
Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
(2017)SAM Research ReportReport -
Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations
(2017)SAM Research ReportReport