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Our focus is on the study of optimal control problem for branching diffusionprocesses. Initially, we introduce the problem in its strong formulation andexpand it to include linearly growing drifts. To ensure its proper definition,we establish bounds on the moments of these processes. We present a relaxedformulation that provides a suitable characterization based on martingalemeasure. We introduce the notion of atomic control and demonstrate theirequivalence to strong controls in the relaxed setting. We establish theequivalence between the strong and relaxed problem, under a Filippov-typeconvexity condition. Furthermore, by defining a control rule, we can restatethe problem as the minimization of a lower semi-continuous function over acompact set, leading to the existence of optimal controls both for the relaxedproblem and the strong one. By utilizing a useful embedding technique, wedemonstrate that the value functions solves a system of HJB equations. This, inturn, leads to the establishment of a verification theorem. We then apply thistheorem to a Linear-Quadratic example and a Kinetic one.

Relaxed formulation for Controlled Branching Diffusions, Existence of an Optimal Control and HJB Equation