Convergence of the deep BSDE method for coupled FBSDEs View Full Text


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Article Info

DATE

2020-07-22

AUTHORS

Jiequn Han, Jihao Long

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. More... »

PAGES

5

References to SciGraph publications

  • 2007. Forward-Backward Stochastic Differential Equations and their Applications in NONE
  • 2004-01. On the Malliavin approach to Monte Carlo approximation of conditional expectations in FINANCE AND STOCHASTICS
  • 2019-03-07. On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations in JOURNAL OF SCIENTIFIC COMPUTING
  • 2017-11-10. Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations in COMMUNICATIONS IN MATHEMATICS AND STATISTICS
  • 1992. Backward stochastic differential equations and quasilinear parabolic partial differential equations in STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS
  • 1999-05. Forward-backward stochastic differential equations and quasilinear parabolic PDEs in PROBABILITY THEORY AND RELATED FIELDS
  • 1989-12. Approximation by superpositions of a sigmoidal function in MATHEMATICS OF CONTROL, SIGNALS, AND SYSTEMS
  • 2019-01-07. Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations in JOURNAL OF NONLINEAR SCIENCE
  • 1994-09. Solving forward-backward stochastic differential equations explicitly — a four step scheme in PROBABILITY THEORY AND RELATED FIELDS
  • 2012-01-21. Least-Squares Monte Carlo for Backward SDEs in NUMERICAL METHODS IN FINANCE
  • 2020-04-06. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations in PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
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    http://scigraph.springernature.com/pub.10.1186/s41546-020-00047-w

    DOI

    http://dx.doi.org/10.1186/s41546-020-00047-w

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