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Controlled Interacting Branching Diffusion Processes: A Viscosity Approach

Antonio Ocello

TL;DR

This work addresses the stochastic control of interacting branching diffusion processes with a measure-valued state and index-dependent dynamics. It builds a dynamic programming framework by bijecting the configuration space with the disjoint union $\bigsqcup_{\mathcal{V}} \mathbb{R}^{d|\mathcal{V}|}$ and derives an infinite system of coupled $HJB$ equations, characterized in the viscosity sense and equipped with a comparison principle. Under natural growth and coercivity assumptions, the value function inherits polynomial bounds and is shown to solve the $HJB$ system; in the mean-field regime a permutation-invariance argument reduces the problem to symmetric controls. The results provide a rigorous foundation for MF control of branching populations and point toward scaling limits to superprocesses and kinetic models.

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.

Controlled Interacting Branching Diffusion Processes: A Viscosity Approach

TL;DR

This work addresses the stochastic control of interacting branching diffusion processes with a measure-valued state and index-dependent dynamics. It builds a dynamic programming framework by bijecting the configuration space with the disjoint union and derives an infinite system of coupled equations, characterized in the viscosity sense and equipped with a comparison principle. Under natural growth and coercivity assumptions, the value function inherits polynomial bounds and is shown to solve the system; in the mean-field regime a permutation-invariance argument reduces the problem to symmetric controls. The results provide a rigorous foundation for MF control of branching populations and point toward scaling limits to superprocesses and kinetic models.

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.
Paper Structure (33 sections, 16 theorems, 176 equations)

This paper contains 33 sections, 16 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Notation
  4. Finite measures.
  5. Label set.
  6. State and control space.
  7. Càdlàg paths.
  8. Branching diffusion processes
  9. Existence and moment estimates.
  10. Control problem
  11. Well-posedness of the control problem.
  12. Differential characterization of the control problem
  13. Dynammic programming principle (DPP)
  14. Verification theorem
  15. Homeomorphisms with $\sqcup_{\mathcal{V}\in{\mathscr{P}_{\mathrm{adm}}(\mathcal{I})}}\mathbb{R}^{d|\mathcal{V}|}$.
  16. ...and 18 more sections

Key Result

Proposition 2.2

Let $(t,\lambda) \in [0,T]\times E$ and $\beta\in\mathcal{S}$. Suppose Assumption HhypH:model_parameters holds. Then, there exists a unique (up to indistinguishability) càdlàg and adapted process $(\xi^{t,\lambda;\beta}_s)_{s\geq t}$ satisfying SDE:strong such that $\xi^{t,\lambda;\beta}_t=\lambda$.

Theorems & Definitions (28)

  • Definition 2.1: Admissible control
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1: DPP
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 18 more