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

2025/11/18 by Tathe, Kartik, Ghosh, Sayan
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.2511.14328

openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28

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.

Citations

• Generalized Space Time Fractional Skellam Process
• Non-Homogeneous Generalized Fractional Skellam Process
• Tempered Fractional Poisson Processes and Fractional Equations with Z-Transform
• The fractional non-homogeneous Poisson process
• A fractional counting process and its connection with the Poisson\n process
• The space-fractional Poisson process

Related

• Generalized Counting Process: its Non-Homogeneous and Time-Changed Versions
• Non-Homogeneous Generalized Fractional Skellam Process
• Fractional non-homogeneous counting process
• On the Multivariate Generalized Counting Process and its Time-Changed Variants
• Limit Theorems for the Fractional Non-homogeneous Poisson Process