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A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen and Philippe von Wurstemberger Memoirs of the American Mathematical Society 284(1410) (2023) https://doi.org/10.1090/memo/1410
A posteriori error analysis and adaptivity for high-dimensional elliptic and parabolic boundary value problems
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
Martin Hutzenthaler, Arnulf Jentzen and von Wurstemberger Wurstemberger Electronic Journal of Probability 25(none) (2020) https://doi.org/10.1214/20-EJP423
Weak Galerkin method with implicit θ-schemes for second-order parabolic problems
Numerical Mathematics and Advanced Applications ENUMATH 2015
Lina von Sydow, Paria Ghafari, Erik Lehto and Mats Wångersjö Lecture Notes in Computational Science and Engineering, Numerical Mathematics and Advanced Applications ENUMATH 2015 112 607 (2016) https://doi.org/10.1007/978-3-319-39929-4_58
A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations
Norbert Hilber, Oleg Reichmann, Christoph Schwab and Christoph Winter Springer Finance, Computational Methods for Quantitative Finance 91 (2013) https://doi.org/10.1007/978-3-642-35401-4_8
Hermite Spectral Method with Hyperbolic Cross Approximations to High-Dimensional Parabolic PDEs
Closed-Form Asymptotics and Numerical Approximations of 1D Parabolic Equations with Applications to Option Pricing
Wen Cheng, Nick Costanzino, John Liechty, Anna Mazzucato and Victor Nistor SIAM Journal on Financial Mathematics 2(1) 901 (2011) https://doi.org/10.1137/100796832
Two-scale finite element discretizations for integro- differential equations
Huajie Chen, Fang Liu, Nils Reich, Christoph Winter and Aihui Zhou Journal of Integral Equations and Applications 23(3) (2011) https://doi.org/10.1216/JIE-2011-23-3-351
Solving chemical master equations by adaptive wavelet compression
Claude Le Bris and Tony Lelièvre Lecture Notes in Computational Science and Engineering, Multiscale Modeling and Simulation in Science 66 49 (2009) https://doi.org/10.1007/978-3-540-88857-4_2
On the solution of the Fokker–Planck equation using a high-order reduced basis approximation
Special Volume: Mathematical Modeling and Numerical Methods in Finance
Olivier Pironneau and Yves Achdou Handbook of Numerical Analysis, Special Volume: Mathematical Modeling and Numerical Methods in Finance 15 369 (2009) https://doi.org/10.1016/S1570-8659(08)00011-2
On the Approximation of Infinite Dimensional Optimal Stopping Problems with Application to Mathematical Finance
Sparse finite element approximation of high-dimensional transport-dominated diffusion problems
Christoph Schwab, Endre Süli and Radu Alexandru Todor ESAIM: Mathematical Modelling and Numerical Analysis 42(5) 777 (2008) https://doi.org/10.1051/m2an:2008027
Sparse adaptive finite elements for radiative transfer
A posteriorierror analysis for parabolic variational inequalities
Kyoung-Sook Moon, Ricardo H. Nochetto, Tobias von Petersdorff and Chen-song Zhang ESAIM: Mathematical Modelling and Numerical Analysis 41(3) 485 (2007) https://doi.org/10.1051/m2an:2007029
Financial Engineering
Liming Feng, Pavlo Kovalov, Vadim Linetsky and Michael Marcozzi Handbooks in Operations Research and Management Science, Financial Engineering 15 301 (2007) https://doi.org/10.1016/S0927-0507(07)15007-6
Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows
Andrea Bonito, Philippe Clément and Marco Picasso ESAIM: Mathematical Modelling and Numerical Analysis 40(4) 785 (2006) https://doi.org/10.1051/m2an:2006030
Linear Complexity Solution of Parabolic Integro-differential Equations