Cause of non-convergence of the deep FBSDE method in the controlled Brownian motion experiment

Ascertain the underlying cause of the discrepancy observed in the two-dimensional controlled Brownian motion experiment with terminal cost g(x) = -|x1 − x2|, where the deep FBSDE (deep BSDE) method minimizes the terminal loss but fails to converge to the correct initial value Y0 and solution, and rigorously characterize the mechanisms responsible for this failure.

Background

The paper studies a controlled Brownian motion problem with T = 0.5, d = 2, r = 1, σ = 0.25, initial state x0 = (−0.1, 0.1)⊤, and terminal cost g(x) = −|x1 − x2|. In this setting, the standard deep FBSDE (deep BSDE) method drives the loss function near zero but produces an incorrect Y0 and qualitatively wrong trajectories, while the proposed deep multi-FBSDE method converges to the correct solution.

The authors highlight that despite achieving a small terminal mismatch, the deep FBSDE method yields dynamics inconsistent with the reference solution derived via the Cole–Hopf transform and Monte Carlo. They explicitly note that the reason for this discrepancy is not currently understood.