• Corpus ID: 246035973

Convergence of a robust deep FBSDE method for stochastic control

@article{Andersson2022ConvergenceOA,
  title={Convergence of a robust deep FBSDE method for stochastic control},
  author={Kristoffer Andersson and Adam Andersson and Cornelis W. Oosterlee},
  journal={ArXiv},
  year={2022},
  volume={abs/2201.06854}
}
In this paper we propose a deep learning based numerical scheme for strongly coupled FBSDE, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the means square error in the terminal condition. We show by a numerical example that a direct extension of the classical… 

Figures and Tables from this paper

Numerical Methods for Backward Stochastic Differential Equations: A Survey
TLDR
This paper focuses on the core features of each method: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, in order to provide an exhaustive up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and useful comparison and categorization.

References

SHOWING 1-10 OF 86 REFERENCES
Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control
A modified Deep BSDE (backward differential equation) learning method with measurability loss, called Deep BSDE-ML method, is introduced in this paper to solve a kind of linear decoupled
Convergence of the deep BSDE method for coupled FBSDEs
TLDR
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.
The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions
A novel discretization is presented for forward-backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity
Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients
TLDR
Under mild conditions, a posterior estimate of the numerical solution that holds for any time duration validates the convergence of the recently proposed Deep BSDE method.
A control method for solving high-dimensional Hamiltonian systems through deep neural networks
TLDR
Compared with the Deep FBSDE method, the novel algorithms converge faster, which means that they require fewer training steps, and demonstrate more stable convergences for different Hamiltonian systems.
Deep backward schemes for high-dimensional nonlinear PDEs
TLDR
The proposed new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs) rely on the classical backward stochastic differential equation (BSDE) representation of PDEs and provide error estimates in terms of the universal approximation of neural networks.
Some machine learning schemes for high-dimensional nonlinear PDEs
TLDR
The proposed new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs) rely on the classical backward stochastic differential equation (BSDE) representation of PDEs and provide error estimates in terms of the universal approximation of neural networks.
Approximation Error Analysis of Some Deep Backward Schemes for Nonlinear PDEs
TLDR
The proposed algorithm, called deep backward multistep scheme (MDBDP), is a machine learning version of the LSMDP scheme and yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme.
Approximate stochastic control based on deep learning and forward backward stochastic differential equations
TLDR
A nonlinear Gaussian diffusion type state equation with control in the drift and a quadratic cost functional with finite time horizon is considered and the proposed algorithm relies on recent work on deep learning based solvers for backward stochastic differential equations.
Deep Learning Approximation for Stochastic Control Problems
TLDR
This work develops a deep learning approach that directly solves high-dimensional stochastic control problems based on Monte-Carlo sampling and approximate the time-dependent controls as feedforward neural networks and stack these networks together through model dynamics.
...
...