Generalization of the SMP-based deep BSDE approach to non-constant control settings

Determine to what extent the deep BSDE approach formulated through a BSDE derived from the stochastic maximum principle generalizes to problems in which the control process depends nontrivially on the state, rather than being constant over the state space.

Background

The paper compares its results with an approach that applies the deep BSDE method to a BSDE obtained from the stochastic maximum principle for the same LQG problem. In that specific instance, the control appearing in the BSDE is constant in the state space, which simplifies the problem considerably.

The authors explicitly state that it is unclear whether that approach would extend to more complex problems where the control is state-dependent, thus highlighting an unresolved generalization question.