Long-Term Average Impulse Control with Mean Field Interactions

TL;DR

The paper tackles long-run ergodic impulse-control problems with mean-field interactions for a one-dimensional diffusion on an interval, where the unit impulse price $p$ depends on the market mean-field via $p=\varphi(z)$ with $z=\kappa^Q$. It develops a tractable, explicit framework based on $(w,y)$-threshold policies and renewal theory to obtain both a mean-field game (MFG) equilibrium and a mean-field control (MFC) optimum, using a fixed-point approach to connect individual incentives to the aggregate supply rate. The main contributions are (i) an existence and explicit characterization of an MFG equilibrium via a fixed-point $z^*=\mathfrak z\circ\Psi\circ\varphi(z^*)$ and $(w^*,y^*)$, (ii) a mean-field control solution yielding an optimal $(w^*,y^*)$ with $z=\mathfrak z(w^*,y^*)$ and an explicit value, and (iii) a detailed comparison showing that the centralized MFC can outperform the decentralized MFG, albeit with reduced robustness. The theoretical results are illustrated through stochastic logistic growth and population-growth models, highlighting practical implications for renewable-resource management and financial decision-making under mean-field interactions.

Abstract

This paper analyzes and explicitly solves a class of long-term average impulse control problems with a specific mean-field interaction. The underlying process is a general one-dimensional diffusion with appropriate boundary behavior. The model is motivated by applications such as the optimal long-term management of renewable resources and financial portfolio management. Each individual agent seeks to maximize her long-term average reward, which consists of a running reward and income from discrete impulses, where the unit intervention price depends on the market through a stationary supply rate, the specific mean field variable to be considered. In a competitive market setting, we establish the existence of and explicitly characterize an equilibrium strategy within a large class of policies under mild conditions. Additionally, we formulate and solve the mean field control problem, in which agents cooperate with each other, aiming to realize a common maximal long-term average profit. To illustrate the theoretical results, we examine a stochastic logistic growth model and a population growth model in a stochastic environment with impulse control.

Theorems & Definitions (40)

• Definition 2.2: Admissibility
• Definition 2.3: $(w,y)$-Policies
• Remark 2.4
• Remark 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• Remark 3.3
• Proposition 3.4
• proof