Long-Term Average Impulse and Singular Control of a Growth Model with Two Revenue Sources

TL;DR

This paper develops an explicit framework for solving long-term average impulse control problems and their singular-control counterparts for a general one-dimensional diffusion with prescribed boundary behavior. By exploiting renewal theory, it derives a fractional objective $F(w,y; p,K,\gamma)$ that governs an $(w,y)$-policy and proves existence and, under suitable conditions, uniqueness of an optimal policy, characterized by the first-order conditions $F^*=h(w^*)=h(y^*)$ with $h(x)=\dfrac{\gamma g'(x) + p}{\xi'(x)}$. The singular-control problem emerges as the $K\to 0$ limit, with the optimal singular policy given by reflections at a level $\widehat{y}$ and the value $\widehat{J}(Z_x)=h(x)$; the paper rigorously quantifies the penalty due to positive fixed costs and shows convergence of impulse policies to the singular solution. A detailed sensitivity analysis and the connection to a probabilistic cell problem and overtaking optimality provide practical insights and methodological tools for applications in renewable-resource management and portfolio optimization. Overall, the results establish a precise link between impulse and singular controls and offer a tractable, explicit approach to long-horizon decision problems under diffusion dynamics.

Abstract

This paper analyzes and explicitly solves a class of long-term average impulse control problems and a related class of singular control problems. 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. A large class of admissible policies is identified over which the agent seeks to maximize her long-term average reward, consisting of a running reward and income from either discrete impulses or singular actions. The long-term expected total reward and its relation to overtaking optimality is also considered. Sensitivity analysis with regard to the parameters of the impulse control model are performed. Key connections between the impulse and singular control problems are displayed.

Figures (1)

• Figure 1: (a) The function $r$. (b) The function $h$.

Theorems & Definitions (52)

• Proposition 2.2
• Definition 2.3: Impulse Admissibility
• Remark 2.4
• Proposition 2.7
• Definition 3.1: $(w,y)$-Policies
• Lemma 3.2
• proof
• Proposition 3.3
• Lemma 3.4
• Lemma 3.5