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

TL;DR

This paper explores martingale characterizations of non-homogeneous counting processes and their fractional variants, establishing new theoretical links and characterizations for these stochastic processes and their time-changed versions.

Contribution

It introduces martingale characterizations for non-homogeneous generalized counting processes and their fractional, time-changed, and Skellam variants, expanding theoretical understanding.

Findings

Martingale characterizations for NGCP are established.

Equivalent forms of martingale characterizations are demonstrated.

Characterizations for time-changed and Skellam variants are provided.

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.