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

K. L. Helmes, R. H. Stockbridge+1 more|arXiv (Cornell University)|Jan 14, 2026

Economic theories and models0 citations

TL;DR

The paper analyzes and explicitly solves long-term average impulse control and related singular control problems for a one-dimensional diffusion with two revenue streams, using renewal theory and quasi-variational inequalities to characterize optimal policies.

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.

Motivation & Objective

• Motivate and model long-term average impulse and singular control problems arising in renewable resource management and financial portfolio contexts.
• Identify a large class of admissible impulse policies and characterize when an $(s,S)$-type (here $(w,y)$) policy is optimal.
• Establish connections between impulse control and singular control, including overtaking optimality considerations.
• Develop a renewal-theoretic framework to express and analyze long-term rewards and supply rates.
• Perform sensitivity analysis with respect to model parameters and clarify the relationship between impulse and singular controls.

Proposed method

• Model the state as a one-dimensional diffusion with boundary behavior as in Condition 2.1.
• Define impulse policies $R=igl\u0303( au_k,Y_k)igr o$ and derive the controlled process $X^R$ (1.2).
• Introduce $(w,y)$-policies and express their payoff via the function $F(w,y;p,K,eta)$ using a renewal argument (3.3).
• Characterize the stationary density under $(w,y)$ and relate cycle length to the potential functions $\xi$ and $g$ (3.4, 3.5).
• Formulate and analyze the long-term average impulse control problem, including existence and uniqueness results for the optimal policy, via quasi-variational inequalities (QVI) (Section 4).
• Explore the singular control problem as the zero-fixed-cost limit and discuss overtaking optimality (Section 7).

Experimental results

Research questions

• RQ1 What are the optimal impulse strategies for maximizing long-term average profit in this diffusion setting?
• RQ2 Under what conditions does a unique $(w,y)$ policy maximize the long-term average reward?
• RQ3 How are the impulse and singular control problems connected, and what is the limit behavior as fixed costs vanish?
• RQ4 How does the stationary distribution and the renewal structure inform the value and optimality of policies?
• RQ5 What is the sensitivity of the optimal policy to parameters $(p,K,eta)$ and the running reward function $c$?

Key findings

• An explicit form for the long-term average payoff of $(w,y)$ policies is derived as F(w,y;p,K, ildeeta).
• The stationary density under a $(w,y)$ policy is given by a piecewise expression involving the speed measure and scale function (3.4).
• A bound z0 on the admissible supply rate is established, with z in (0,z0) attainable by some $(w,y)$ policy (Proposition 3.3).
• Under appropriate conditions, there exists a unique optimal $(w,y)$ policy and the function G (the impulse reward potential) along with F* solve the associated QVI (Section 4).
• The impulse control solution converges to the singular control solution as K → 0, linking the two problems (Section 7).
• Sensitivity analysis shows how the optimal policy and value respond to $(p,K,eta)$ and to the running reward c (Section 6).