Optimal control and filtering of the reproduction law of a branching process

A finite horizon control problem for the reproduction law of a branching pr ocess is studied. Some examples with complete information are tackled via t he Hamilton-Jacobi-Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and th e observations and the observation process has unbounded intensity, a Girsa nov-type change of probability measure can be set and the filtering equatio n for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, a lso for the Kushner-Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitti ng process and the unnormalized conditional distribution. By the strong uni queness for the Zakai equation, equivalence between the partially observabl e control problem and the separated one is proved.