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Laplace Transforms of Stopping Times for Subordinator with Applications to Inventory Control

Ryoya Koide

Abstract

Intermittent demand fluctuations pose significant challenges in disaster logistics and medical supply systems. In this study, we formulate cumulative demand as a generalized Lévy process composed of a drift term, Poisson jumps, and compound Poisson jumps, and analyze a continuous-time inventory model. The proposed framework provides a unified formulation that encompasses both drifted Poisson processes and drifted compound Poisson processes. From a mathematical perspective, we treat the reorder time as a first-passage problem of a subordinator and derive its Laplace transform via the Laplace exponent. In particular, for the drifted Poisson case, we obtain an explicit representation of the inverse Laplace exponent using the Lambert W function, which yields an analytic expression for the Laplace transform of the first-passage time. Furthermore, when the jump sizes follow exponential and Gamma distributions, we derive explicit formulas for the mean and variance of the reorder times, thereby clarifying the moment structure of first-passage times for generalized Lévy demand processes. From an operations research perspective, we explicitly characterize the expected total cost over a finite time horizon based on the distribution of cumulative demand. This study presents an analytical framework that integrates first-passage theory of Lévy processes with continuous-time inventory control.

Laplace Transforms of Stopping Times for Subordinator with Applications to Inventory Control

Abstract

Intermittent demand fluctuations pose significant challenges in disaster logistics and medical supply systems. In this study, we formulate cumulative demand as a generalized Lévy process composed of a drift term, Poisson jumps, and compound Poisson jumps, and analyze a continuous-time inventory model. The proposed framework provides a unified formulation that encompasses both drifted Poisson processes and drifted compound Poisson processes. From a mathematical perspective, we treat the reorder time as a first-passage problem of a subordinator and derive its Laplace transform via the Laplace exponent. In particular, for the drifted Poisson case, we obtain an explicit representation of the inverse Laplace exponent using the Lambert W function, which yields an analytic expression for the Laplace transform of the first-passage time. Furthermore, when the jump sizes follow exponential and Gamma distributions, we derive explicit formulas for the mean and variance of the reorder times, thereby clarifying the moment structure of first-passage times for generalized Lévy demand processes. From an operations research perspective, we explicitly characterize the expected total cost over a finite time horizon based on the distribution of cumulative demand. This study presents an analytical framework that integrates first-passage theory of Lévy processes with continuous-time inventory control.
Paper Structure (9 sections, 14 theorems, 135 equations)

This paper contains 9 sections, 14 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Fixed-Order-Quantity Policy under Intermittent Demand
  4. Expected Total Cost
  5. Main Results
  6. Case where the Cumulative Demand is a Drifted Poisson Process
  7. Case where the Cumulative Demand is a Drifted Compound Poisson Process
  8. Generalization of the Cumulative Demand
  9. On the Expected Total Cost

Key Result

Theorem 1

Let $\{X_t\}$ be a subordinator (a one-dimensional, possibly killed, non-decreasing Lévy process), and define Let $\Psi(\theta)= \log E[\exp(\theta X_1)]$ denote the Laplace exponent. Then,

Theorems & Definitions (33)

  • Theorem : cf. J.Bertoin (1996) bertoin1996levy, Sato (1999) sato1999levy
  • Definition 2.1: First Passage Time of $\{D_t\}$
  • Definition 2.2: Inventory Level Process
  • Definition 2.3: Expected Inventory Cost
  • Remark 2.4
  • Definition 3.1: Cumulative Demand
  • Definition 3.2: Lambert W function MR1414285
  • Lemma 3.3: Derivative of the Lambert W functionMR1414285
  • Theorem 3.4
  • proof
  • ...and 23 more