Controlled superprocesses and HJB equation in the space of finite measures

Abstract

This paper introduces the formalism required to analyze a certain class of stochastic control problems that involve a super diffusion as the underlying controlled system. To establish the existence of these processes, we show that they are weak scaling limits of controlled branching processes. First, we prove a generalized Itô's formula for this dynamics in the space of finite measures, using the differentiation in the space of finite positive measures. This lays the groundwork for a PDE characterization of the value function of a control problem, which leads to a verification theorem. Finally, focusing on an exponential-type value function, we show how a regular solution to a finite--dimensional HJB equation can be used to construct a smooth solution to the HJB equation in the space of finite measures, via the so-called branching property technique.

References in corpus (6)

• Mean field games via controlled martingale problems: Existence of Markovian equilibria
• Kolmogorov equations on spaces of measures associated to nonlinear filtering processes
• A Stochastic Target Problem for Branching Diffusions
• Itô's formula for flows of measures on semimartingales
• Splitting methods and short time existence for the master equations in mean field games
• Controlled Interacting Branching Diffusion Processes: Relaxed Formulation in the Mean-Field Regime

Cited by in corpus (1)

• Optimal Stopping of Branching Diffusion Processes