Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations

Abstract

In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.

28 pages, 16 figures, 10 tables

References in corpus (9)

• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Time discretization and Markovian iteration for coupled FBSDEs
• Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver
• Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients
• Towards Robust and Stable Deep Learning Algorithms for Forward Backward Stochastic Differential Equations
• Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations