Optimal control of branching diffusion processes: a finite horizon problem

TL;DR

This paper develops a control theory for branching diffusion processes, showing the value function solves a nonlinear PDE, and extends dynamic programming to handle dependencies in the process.

Contribution

It introduces a novel approach to optimal control of branching diffusions, including a new dynamic programming principle and PDE characterization of the value function.

Findings

Value function is the unique viscosity solution of a nonlinear PDE.

Extended dynamic programming approach for dependent branching processes.

Provided a rigorous formulation and justification of the control problem.

Abstract

In this paper, we aim to develop the theory of optimal stochastic control for branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing a criterion that is expressed as the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to recover a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.