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.
Highlights
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 the linear case, Feynman–Kac formula and Monte Carlo simulation together provide an efficient approach to solving PDEs and associated Backward stochastic differential equation forward-backward stochastic differential equations (FBSDEs) (BSDE) numerically
Overall there is no numerical algorithm in literature so far proved to overcome the curse of dimensionality for general quasilinear parabolic PDEs and the corresponding FBSDEs
Summary
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. 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 (Pardoux and Peng 1992), the algorithms designed for one kind of equation can often be used to solve another one. Thanks to the universal approximation capability and parsimonious parameterization of neural networks, in practice the objective function can be effectively optimized in high-dimensional cases, and the function values of interests are obtained quite accurately. To the best of our knowledge, this is the first theoretical result of the deep BSDE method for solving FBSDEs and parabolic PDEs. our numerical algorithm is based on neural networks, the theoretical result provided here is applicable to the algorithms based on other forms of function approximations.
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