Optimal Control of a Levy Inventory System: The Optimality of Control Band Policy

TL;DR

This paper models an inventory system using Levy processes, formulates an impulse control problem, and proves the optimality of a simple control band policy with analytical solutions and behavior analysis.

Contribution

It establishes the existence and closed-form solution of the optimal control, proving the optimality of a control band policy for Levy inventory systems.

Findings

Optimal control exists and is characterized in closed form.

Control band policy is proven to be optimal.

Analyzes transient and steady-state behaviors of the controlled process.

Abstract

We consider an inventory system whose state is modeled by a L\'{e}vy process. There are two types of costs--the running costs and the inventory control costs. The running costs (also known as the holding/penalty costs) are incurred continuously at some rate as a function of the inventory state. The inventory control costs, incurred only when interventions of the inventory state are placed, have both a fixed and a variable component. The objective is to minimize the expectation of the infinite horizon discounted costs. We formulate this as a stochastic impulse control problem. In our setting, we obtain analytical results that are of significant implications. Specifically, we establish the existence of the optimal control, and we provide the solution in closed-form. More importantly, we prove the optimality of the simple control band policy. Furthermore, we investigate the transient and the steady-state behavior of the controlled process and the stochastic decomposition property.