Convergence of a robust deep FBSDE method for stochastic control

Journal article, 2023

In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming
from stochastic control. It is a modification of the deep BSDE method in which the initial value to the
backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost
of the control problem, and a variance term which coincides with the mean squared error in the terminal
condition. We show by a numerical example that a direct extension of the classical deep BSDE method
to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works.
Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete
control problems, we provide an error analysis for our method. We show empirically that the method
converges for three different problems, one being the one that failed for a direct extension of the deep BSDE
method.