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Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration con…
Pricing and hedging American-style options with deep learning
Sebastian Becker, Patrick Cheridito, Arnulf Jentzen
In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a low…
Efficient approximation of high-dimensional functions with neural networks
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach i…
Generalised multilevel Picard approximations
Michael B. Giles, Arnulf Jentzen, Timo Welti
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numeri…
Uniform error estimates for artificial neural network approximations for heat equations
Lukas Gonon, Philipp Grohs, Arnulf Jentzen +2
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly r…
Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations
Martin Hutzenthaler, Arnulf Jentzen, Felix Lindner +1
The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochas…