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.

arXiv comment28 pages, 16 figures, 10 tables

References in corpus (21)

• Adam: A Method for Stochastic Optimization
• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• Solving high-dimensional partial differential equations using deep learning
• 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
• Convergence of the Deep BSDE Method for Coupled FBSDEs
• On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
• Deep splitting method for parabolic PDEs
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• Time discretization and Markovian iteration for coupled FBSDEs
• Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
• Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
• Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• 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
• Multistep schemes for solving backward stochastic differential equations on GPU
• A Review of Tree-based Approaches to solve Forward-Backward Stochastic Differential Equations
• Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations