Optimal Control of McKean--Vlasov Branching Diffusion Processes

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.