Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

Abstract

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

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Acknowledgements

Sebastian Becker and Jiequn Han are gratefully acknowledged for their helpful and inspiring comments regarding the implementation of the deep 2BSDE method.

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Appendix: Source Codes

Appendix: Source Codes

A.1: A Python Code for the Deep 2BSDE Method used in Subsection 4.1

The following Python code, Python code 1, is a simplified version of Python code 3 in Appendix A.3.

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A.2: A Matlab Code for the Branching Diffusion Method used in Subsection 4.1

The following Matlab code is a slightly modified version of the Matlab code in (E et al. (2017c), Subsection 6.2).

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A.3: A Python Code for the Deep 2BSDE Method used in Subsection 4.3

The following Python code is based on the Python code in (E et al. (2017c), Subsection 6.1).

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A.4: A Matlab Code for the Classical Monte Carlo Method used in Subsection 4.4

The following Matlab code is a slightly modified version of the Matlab code in (E et al. (2017c), Subsection 6.3).

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A.5: A Matlab Code for the Finite Differences Method used in Subsection 4.6

The following Matlab code is inspired by the Matlab code in (E et al. (2017b), MATLAB code 7 in Section 3).

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Beck, C., E, W. & Jentzen, A. Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations. J Nonlinear Sci 29, 1563–1619 (2019). https://doi.org/10.1007/s00332-018-9525-3

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Keywords

  • Deep learning
  • Second-order backward stochastic differential equation
  • 2BSDE
  • Numerical method
  • Black–Scholes–Barenblatt equation
  • Knightian uncertainty
  • Hamiltonian–Jacobi–Bellman equation
  • HJB equation
  • Nonlinear expectation
  • G-Brownian motion