Optimal Control of McKean--Vlasov Branching Diffusion Processes
Julien Claisse, Jiazhi Kang, Tianxu Lan, Xiaolu Tan
TL;DR
The paper develops a rigorous framework for optimal control of McKean–Vlasov branching diffusion processes, where interactions are mediated by the marginal alive-particle measure. It establishes existence and uniqueness of the controlled branching SDE, proves a dynamic programming principle, and derives an HJB master equation on the space of finite measures, complemented by a verification theorem. Under suitable regularity, it demonstrates that the value function solves the master equation, and it provides an explicit linear–quadratic example whose solution is characterized by Riccati-type ODEs, including a closed-form optimal control. This work advances mean-field control with branching dynamics by connecting a nonlinear Fokker–Planck equation, a measure-valued HJB master equation, and tractable LQ solutions.
Abstract
We study an optimal control problem of McKean--Vlasov branching diffusion processes, in which the interaction term is determined by the marginal measure induced by all alive particles in the system. Accordingly, the value function is defined on the space of finite nonnegative measures over the Euclidean space. Within the framework of Lipschitz continuous closed-loop controls, and by using the uniqueness of solution to the associated nonlinear Fokker--Planck equation, we establish the dynamic programming principle. Further, under the regularity assumptions, we show that the value function satisfies a Hamilton--Jacobi--Bellman (HJB) master equation defined on the space of finite nonnegative measures. We next provide a corresponding verification theorem. Finally, we study a linear--quadratic controlled branching processes problem, for which explicit solutions are derived in terms of Riccati-type equations.