The Optimal Branching Problem under Branching Diffusion Model

In this paper, the optimal branching times of a class of diffusion branching processes are studied systematically. The criterion considered is risk-sensitive. In order to solve this kind of problem, we first study the optimal branching times of the branching Feller process and develop the corresponding dynamic programming principle with the aid of the martingale method. At the same time, we also give the characterization of the optimal branching times. It is shown that the value function of the original problem is the unique viscosity solution of the corresponding variational inequality. It should be pointed out that under the framework of our proof, the verification of the existence of viscosity solutions becomes very concise. An example for the main result is given and the optimal branching times are constructed.