Convergence of a robust deep FBSDE method for stochastic control
This work addresses a specific numerical challenge in stochastic control for researchers in computational finance and applied mathematics, representing an incremental improvement over existing methods.
The authors tackled the failure of directly extending the deep BSDE method to solve forward-backward stochastic differential equations (FBSDEs) from stochastic control by proposing a modified deep learning scheme with a new loss function and fixed initial value, showing empirical convergence for three problems including one where the direct extension failed.
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.