Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

Abstract

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

39 pages, 15 figures

References in corpus (6)

• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
• Deep Learning Approximation for Stochastic Control Problems
• Counterparty Risk Valuation: A Marked Branching Diffusion Approach

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