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Martingale Characterizations of Non-Homogeneous Counting Processes and Their Fractional Variants

Kartik Tathe, Sayan Ghosh

TL;DR

The paper develops a comprehensive martingale framework for non-homogeneous counting processes and their fractional generalizations. It proves that a non-homogeneous generalized counting process (NGCP) can be represented as a weighted sum of non-homogeneous Poisson processes (NPPs) and that compensated and exponential martingale forms are interchangeable. The authors extend these characterizations to a wide family of time-changed and fractional variants, using stable, tempered stable, and mixed subordinators, and they derive analogous results for Skellam-type processes. This unified treatment provides rigorous tools for modeling non-Markovian counting phenomena in applied fields such as queueing, finance, and reliability, and clarifies the structural connections between classical and fractional counting models.

Abstract

This paper investigates the martingale characterizations of non-homogeneous counting processes and their fractional generalizations. We show that the weighted sum of non-homogeneous Poisson processes (NPPs) is the non-homogeneous generalized counting process (NGCP). Both the compensated and exponential forms of martingale characterization for NGCP are obtained, and are shown to be equivalent. Moreover, we provide martingale characterizations for various time-changed variants of the NGCP and their Skellam versions using stable and/or inverse stable subordinators.

Martingale Characterizations of Non-Homogeneous Counting Processes and Their Fractional Variants

TL;DR

The paper develops a comprehensive martingale framework for non-homogeneous counting processes and their fractional generalizations. It proves that a non-homogeneous generalized counting process (NGCP) can be represented as a weighted sum of non-homogeneous Poisson processes (NPPs) and that compensated and exponential martingale forms are interchangeable. The authors extend these characterizations to a wide family of time-changed and fractional variants, using stable, tempered stable, and mixed subordinators, and they derive analogous results for Skellam-type processes. This unified treatment provides rigorous tools for modeling non-Markovian counting phenomena in applied fields such as queueing, finance, and reliability, and clarifies the structural connections between classical and fractional counting models.

Abstract

This paper investigates the martingale characterizations of non-homogeneous counting processes and their fractional generalizations. We show that the weighted sum of non-homogeneous Poisson processes (NPPs) is the non-homogeneous generalized counting process (NGCP). Both the compensated and exponential forms of martingale characterization for NGCP are obtained, and are shown to be equivalent. Moreover, we provide martingale characterizations for various time-changed variants of the NGCP and their Skellam versions using stable and/or inverse stable subordinators.

Paper Structure

This paper contains 13 sections, 6 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Martingale
  4. Non-homogeneous Poisson process
  5. Non-homogeneous generalized counting process
  6. Stable subordinator and its inverse
  7. Tempered stable subordinator
  8. Mixed stable subordinator and its inverse
  9. Martingale characterizations of non-homogeneous counting processes
  10. Martingale characterization of NPP
  11. Equivalence of martingales
  12. Martingale characterization of NGCP
  13. Martingale characterizations of fractional variants

Key Result

Lemma 3.1

Let $\{\mathcal{N}(t)\}_{t\ge 0}$ be a counting process with jump size $+1$ with rate $\lambda(t)$ such that $\mathcal{N}(0)=0$ and $\mathcal{F}_t=\sigma(\{\mathcal{N}(s)\},\,s\le t)$, $t\ge 0$ be a filtration generated by it. Then $\{X(\mathcal{N}(t),t)=\exp\{u\mathcal{N}(t)-(e^u-1)\Lambda(t)\}\}_{

Theorems & Definitions (25)

  • Lemma 3.1
  • proof
  • proof
  • Proposition 3.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 15 more