Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1563–1619 | Cite as

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

  • Christian BeckEmail author
  • Weinan E
  • Arnulf Jentzen
Original Paper


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.


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 



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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Authors and Affiliations

  1. 1.ETH ZurichZurichSwitzerland
  2. 2.Princeton UniversityPrincetonUSA
  3. 3.Beijing Institute of Big Data ResearchBeijingChina

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