Optimal Control of Mean-field FBSDEs and Classical Solutions of the Related Generalized HJB Equations
Abstract
In this article, we study optimal control problems driven by mean-field forward-backward stochastic differential equations (FBSDEs). The aim is two-fold: (i) to provide the unique optimal control; (ii) to establish classical solutions of the associated generalized Hamilton-Jacobi-Bellman (HJB) equations on the space of probability measures. For (i), we study the coupled system arising from the maximum principle, which involves both the state FBSDEs and the adjoint FBSDEs. This system is non-standard and more complicated than that in standard mean-field type control (MFTC) problems. By extending the method of continuation in the coefficients to our framework, we establish the well-posedness result, and subsequently obtain the differentiability of the value functional with respect to the distribution argument --- thereby extending existing results for MFTC problems. For (ii), since the flow property does not generally hold for control problems of FBSDEs, standard dynamic-programming arguments cannot be applied directly. We therefore develop probabilistic approaches to the classical solvability of generalized HJB equations under three distinct sets of conditions. The auxiliary processes arising in these arguments differ significantly from those in standard MFTC problems. We establish new connections between the value function and the adjoint processes, derive equations for the relevant decoupling field, and obtain first- and second-order generalized HJB equations, including a coupled system on the space of probability measures. The standard HJB equations for MFTC problems are recovered as special cases. Our approach yields classical solutions without requiring non-degeneracy of the diffusion.
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BibTeX
@misc{huang2026optimal,
title = {{Optimal Control of Mean-field FBSDEs and Classical Solutions of the Related Generalized HJB Equations}},
author = {Ziyu Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18015},
eprint = {2610.18015},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18015},
}
RIS
TY - UNPB
TI - Optimal Control of Mean-field FBSDEs and Classical Solutions of the Related Generalized HJB Equations
AU - Ziyu Huang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18015
UR - https://arxiv.org/abs/2610.18015
AB - In this article, we study optimal control problems driven by mean-field forward-backward stochastic differential equations (FBSDEs). The aim is two-fold: (i) to provide the unique optimal control; (ii) to establish classical solutions of the associated generalized Hamilton-Jacobi-Bellman (HJB) equations on the space of probability measures. For (i), we study the coupled system arising from the maximum principle, which involves both the state FBSDEs and the adjoint FBSDEs. This system is non-standard and more complicated than that in standard mean-field type control (MFTC) problems. By extending the method of continuation in the coefficients to our framework, we establish the well-posedness result, and subsequently obtain the differentiability of the value functional with respect to the distribution argument --- thereby extending existing results for MFTC problems. For (ii), since the flow property does not generally hold for control problems of FBSDEs, standard dynamic-programming arguments cannot be applied directly. We therefore develop probabilistic approaches to the classical solvability of generalized HJB equations under three distinct sets of conditions. The auxiliary processes arising in these arguments differ significantly from those in standard MFTC problems. We establish new connections between the value function and the adjoint processes, derive equations for the relevant decoupling field, and obtain first- and second-order generalized HJB equations, including a coupled system on the space of probability measures. The standard HJB equations for MFTC problems are recovered as special cases. Our approach yields classical solutions without requiring non-degeneracy of the diffusion.
ER -