443 citations · 2.4k across the 59 of their papers we have counts for
59 papers
Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem
Martin Hutzenthaler, Tuan Anh Nguyen
Full history recursive multilevel Picard (MLP) approximations have been proved to overcome the curse of dimensionality in the numerical approximation of semilinear heat equations w…
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Katharina Pohl
The classical Feynman-Kac identity represents solutions of linear partial differential equations in terms of stochastic differential euqations. This representation has been general…
On existence and uniqueness properties for solutions of stochastic fixed point equations with gradient-dependent nonlinearities
Katharina Pohl, Martin Hutzenthaler
The combination of the Itô formula and the Bismut-Elworthy-Li formula implies that suitable smooth solutions of semilinear Kolmogorov partial differential equations (PDEs) are also…
Strong convergence rates for full-discrete approximations of stochastic Burgers equations with multiplicative noise
Martin Hutzenthaler, Robert Link
In this article we establish strong convergence rates on the whole probability space for explicit full-discrete approximations of stochastic Burgers equations with multiplicative t…
A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations
Martin Hutzenthaler, Tuan Anh Nguyen
We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with…
Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations
Petru A. Cioica-Licht, Martin Hutzenthaler, P. Tobias Werner
We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, whi…