Gradient Convergence of Deep Learning-Based Numerical Methods for BSDEs

The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations (SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method, quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K-1/4) rate of gradient convergence with K being the total number of iterative steps.