Controlled Interacting Branching Diffusion Processes: A Viscosity Approach

TL;DR

This paper investigates optimal control of interacting branching diffusion processes using viscosity solutions, establishing a connection with coupled HJB equations and simplifying in the mean-field regime.

Contribution

It introduces a rigorous viscosity framework for the control of measure-valued branching diffusions and characterizes the value function via comparison principles.

Findings

Established a viscosity characterization of the value function.

Proved a comparison principle for the coupled HJB system.

Simplified the control problem in the mean-field symmetric case.

Abstract

We study optimal control problems for interacting branching diffusion processes, a class of measure-valued dynamics capturing both spatial motion and branching mechanisms. From the perspective of the dynamic programming principle, we establish a rigorous connection between the control problem and an infinite system of coupled Hamilton--Jacobi--Bellman (HJB) equations, obtained through a bijection between admissible particle configurations and the disjoint topological union of countable Euclidean spaces. Under natural coercivity conditions on the cost functionals, we show that these growth conditions transfer to the value function and yield a viscosity characterization in the class of functions satisfying the same bounds. We further prove a comparison principle, which allows us to fully characterize the control problem through the associated HJB equation. Finally, we show that the problem simplifies in the mean-field regime, where the model coefficients exhibit symmetry with respect to the indices of the individuals in the population. This permutation invariance allows us to restrict attention to a reduced class of symmetric admissible controls, a reduction established by combining the viscosity characterization of the value function with measurable selection arguments.