An It\^o-type formula for some measure-valued processes and its application on controlled superprocesses

We derive an It\^o-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of It\^o's formula. To address the new challenges arising from the measure-valued setting, we employ symmetric polynomials to approximate the second-order linear derivative of the functional on finite measures, alongside certain localization techniques. A controlled superprocess with a binary branching mechanism can be interpreted as a weak solution to a controlled stochastic partial differential equation (SPDE), which naturally leads to such a decomposition. Consequently, this It\^o-type formula makes it possible to derive the Hamilton-Jacobi-Bellman (HJB) equation and the verification theorem for controlled superprocesses with a binary branching mechanism. Additionally, we propose a heuristic definition for the viscosity solution of an equation involving derivatives on finite measures. We prove that a continuous value function is a viscosity solution in this sense and demonstrate the uniqueness of the viscosity solution when the second-order derivative term on the measure vanishes.