  This article explores an optimal stopping problem for branching diffusion
processes. It consists in looking for optimal stopping lines, a type of
stopping time that maintains the branching structure of the processes under
analysis. By using a dynamic programming approach, we characterize the value
function for a multiplicative cost that depends on the particle's label. We
reduce the problem's dimensionality by setting a branching property and
defining the problem in a finite-dimensional context. Within this framework, we
focus on the value function, establishing polynomial growth and local Lipschitz
properties, together with an innovative dynamic programming principle. This
outcome leads to an analytical characterization with the help of a nonlinear
elliptic PDE. We conclude by showing that the value function serves as the
unique viscosity solution for this PDE, generalizing the comparison principle
to this setting.
