Long-Term Average Impulse Control with Mean Field Interactions

TL;DR

This paper explicitly solves long-term average impulse control problems with mean-field interactions, providing equilibrium strategies and analyzing models relevant to resource management and finance.

Contribution

It introduces explicit solutions and equilibrium characterizations for mean-field impulse control problems with long-term average criteria.

Findings

Explicit equilibrium strategies derived for mean-field impulse control.

Existence of solutions under mild conditions.

Application to logistic and population growth models.

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.