  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.
