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In this work, we propose a new deep learning-based scheme for solving highdimensional nonlinear backward stochastic differential equations (BSDEs). Theidea is to reformulate the problem as a global optimization, where the localloss functions are included. Essentially, we approximate the unknown solutionof a BSDE using a deep neural network and its gradient with automaticdifferentiation. The approximations are performed by globally minimizing thequadratic local loss function defined at each time step, which always includesthe terminal condition. This kind of loss functions are obtained by iteratingthe Euler discretization of the time integrals with the terminal condition. Ourformulation can prompt the stochastic gradient descent algorithm not only totake the accuracy at each time layer into account, but also converge to a goodlocal minima. In order to demonstrate performances of our algorithm, severalhigh-dimensional nonlinear BSDEs including pricing problems in finance areprovided.

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