  In this paper, we aim to develop the theory of optimal stochastic control for
branching diffusion processes where both the movement and the reproduction of
the particles depend on the control. More precisely, we study the problem of
minimizing a criterion that is expressed as the expected value of the product
of individual costs penalizing the final position of each particle. In this
setting, we show that the value function is the unique viscosity solution of a
nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation
corresponding to the problem. To this end, we extend the dynamic programming
approach initiated by Nisio to deal with the lack of independence between the
particles as well as between the reproduction and the movement of each
particle. In particular, we exploit the particular form of the optimization
criterion to recover a weak form of the branching property. In addition, we
provide a precise formulation and a detailed justification of the adequate
dynamic programming principle.
