Convergence of a robust deep FBSDE method for stochastic control

TL;DR

This paper introduces a modified deep learning approach for solving strongly coupled FBSDEs in stochastic control, addressing previous limitations and demonstrating convergence through numerical examples.

Contribution

It presents a novel deep FBSDE method with a new loss function, improving convergence for complex stochastic control problems.

Findings

The new method converges on three different problems.

It overcomes failure of classical deep BSDE in a linear-quadratic case.

Provides error analysis under regularity assumptions.

Abstract

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.