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Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
(2021)Partial Differential Equations and ApplicationsWe introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of ...Journal Article -
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations
(2019)Journal of Nonlinear ScienceHigh-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear ...Journal Article -
Solving high-dimensional partial differential equations using deep learning
(2018)Proceedings of the National Academy of Sciences of the United States of AmericaJournal Article -
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations
(2017)Communications in Mathematics and StatisticsJournal Article