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Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
Sebastian Becker, Benjamin Gess, Arnulf Jentzen +1
The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but,…
Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
Arnulf Jentzen, Diyora Salimova, Timo Welti
In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPD…
Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
Christian Beck, Weinan E, Arnulf Jentzen
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (C…
On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
Weinan E, Martin Hutzenthaler, Arnulf Jentzen +1
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineerin…
Solving high-dimensional partial differential equations using deep learning
Jiequn Han, Arnulf Jentzen, Weinan E
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult pr…
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
Weinan E, Jiequn Han, Arnulf Jentzen
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analog…