W-BSDEs with generators depending on the conditional law of the gradient

Develop a theory of Wasserstein backward stochastic differential equations whose generator depends on the conditional law of the gradient field, and derive a direct characterization of the common-noise-adapted McKean–Vlasov control value without imposing the stated fixed-point condition on the control.

Background

The paper considers a McKean–Vlasov control problem in which the controller is adapted only to the common noise. The proposed optimality condition involves a fixed point for the control, expressed through the conditional mean of the Wasserstein gradient field. The authors explicitly note that this fixed-point condition is not addressed in the paper.

A direct characterization of the corresponding value function would require extending the W-BSDE framework to generators that depend on the conditional law of the variable Z itself. Constructing such an extension is therefore left unresolved.

References

The consistency condition on $\alpha\circ$ is a fixed point, which we do not address here. A direct characterization of $V\circ_{\rm MFC}$, without the fixed-point condition, would require a theory of W-BSDEs whose generator depends on the conditional law of $Z$. We leave this extension for future work.

— W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences  (2610.06593 - Djete et al., 5 Oct 2026) in Remark following Proposition in Section 3.1 (W-BSDE and optimal control of McKean–Vlasov equations with common noise)