Convergence of the deep BSDE method for coupled FBSDEs
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(2020) 5:5
RESEARCH
Open Access
Convergence of the deep BSDE method for coupled FBSDEs Jiequn Han · Jihao Long Received: 26 September 2019 / Accepted: 16 April 2020 / © The Author(s). 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Abstract The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs. Keywords Forward-backward SDE · Weakly coupled condition · Deep learning · High dimension · Numerics
Abbreviations PDE: Partial differential equation SDE: Stochastic differential equation BSDE: Backward stochastic differential equation FBSDE: Forward-backward stochastic differential equation
J. Han () Department of Mathematics, Princeton University, Princeton 08544, NJ, USA e-mail: [email protected] J. Long School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
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J. Han and J. Long
1 Introduction Forward-backward stochastic differential equations (FBSDEs) and partial differential equations (PDEs) of parabolic type have found numerous applications in stochastic control, finance, physics, etc., as a ubiquitous modeling tool. In most situations encountered in practice the equations cannot be solved analytically but require certain numerical algorithms to provide approximate solutions. On the one hand, the dominant choices of numerical algorithms for PDEs are mesh-based methods, such as finite differences, finite elements, etc. On the other hand, FBSDEs can be tackled directly through probabilistic means, with appropriate methods for the approximation of conditional expectation. Since these two kinds of equations are intimately connected through the nonlinear Feynman–Kac formula
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