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Generalized Space Time Fractional Skellam Process

Kartik Tathe, Sayan Ghosh

TL;DR

This work develops the Generalized Space Time Fractional Skellam Process (GSTFSP) and its Skellam counterpart (GSFSP) by combining generalized counting processes with space-time fractional subordination. The authors derive distributional properties, including p.m.f.s, p.g.f.s, and fractional-order moments, and establish governing equations and recurrence relations for these processes. They also provide transition probabilities, arrival and first passage time analyses, increment-process structures, tail asymptotics, limiting laws, infinite-divisibility considerations, and weighted-sum representations, along with martingale characterizations. A running-average framework and Monte Carlo schemes enable simulation and practical path generation, with illustrative plots of pmf, sample paths, and running averages, highlighting the models’ flexibility for modeling complex arrival dynamics in fields such as finance and queuing. Overall, the GSTFSP/GSFSP provide a comprehensive, analytically tractable framework for space-time fractional Skellam-type processes and their variants, including tempered and order-k extensions, with exact representations and asymptotic behavior useful for applications and inference.

Abstract

This paper introduces the Generalized Space-Time Fractional Skellam Process (GSTFSP) and the Generalized Space Fractional Skellam Process (GSFSP). We investigate their distributional properties including the probability generating function (p.g.f.), probability mass function (p.m.f.), fractional moments, mean, variance, and covariance. The governing state differential equations for these processes are derived, and their increment processes are examined. We establish recurrence relations for the state probabilities of GSFSP and related processes. Furthermore, we obtain the transition probabilities, $n^{th}$-arrival times, and first passage times of these processes. The asymptotic behavior of the tail probabilities is analyzed, and limiting distributions as well as infinite divisibility of GSTFSP and GSFSP are studied. We provide the weighted sum representations for these processes and derive their characterizations. Also, we establish the martingale characterization for GSTFSP, GSFSP and related processes. In addition, we introduce the running average processes of GSFSP and its special cases, and obtain their compound Poisson representations. Finally, the p.m.f. of GSTFSP and simulated sample paths for GSTFSP and GSFSP are plotted.

Generalized Space Time Fractional Skellam Process

TL;DR

This work develops the Generalized Space Time Fractional Skellam Process (GSTFSP) and its Skellam counterpart (GSFSP) by combining generalized counting processes with space-time fractional subordination. The authors derive distributional properties, including p.m.f.s, p.g.f.s, and fractional-order moments, and establish governing equations and recurrence relations for these processes. They also provide transition probabilities, arrival and first passage time analyses, increment-process structures, tail asymptotics, limiting laws, infinite-divisibility considerations, and weighted-sum representations, along with martingale characterizations. A running-average framework and Monte Carlo schemes enable simulation and practical path generation, with illustrative plots of pmf, sample paths, and running averages, highlighting the models’ flexibility for modeling complex arrival dynamics in fields such as finance and queuing. Overall, the GSTFSP/GSFSP provide a comprehensive, analytically tractable framework for space-time fractional Skellam-type processes and their variants, including tempered and order-k extensions, with exact representations and asymptotic behavior useful for applications and inference.

Abstract

This paper introduces the Generalized Space-Time Fractional Skellam Process (GSTFSP) and the Generalized Space Fractional Skellam Process (GSFSP). We investigate their distributional properties including the probability generating function (p.g.f.), probability mass function (p.m.f.), fractional moments, mean, variance, and covariance. The governing state differential equations for these processes are derived, and their increment processes are examined. We establish recurrence relations for the state probabilities of GSFSP and related processes. Furthermore, we obtain the transition probabilities, -arrival times, and first passage times of these processes. The asymptotic behavior of the tail probabilities is analyzed, and limiting distributions as well as infinite divisibility of GSTFSP and GSFSP are studied. We provide the weighted sum representations for these processes and derive their characterizations. Also, we establish the martingale characterization for GSTFSP, GSFSP and related processes. In addition, we introduce the running average processes of GSFSP and its special cases, and obtain their compound Poisson representations. Finally, the p.m.f. of GSTFSP and simulated sample paths for GSTFSP and GSFSP are plotted.

Paper Structure

This paper contains 28 sections, 19 theorems, 161 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable subordinator and its inverse
  4. Tempered Stable Subordinator and its inverse
  5. Multinomial Theorem
  6. Caputo–Djrbashian fractional derivative
  7. Lemma
  8. GSTFCP p.m.f.
  9. Two-parameter Mittag-Leffler function
  10. Generalized Space Time Fractional Skellam process
  11. Probability mass function of GSTFSP
  12. Probability Generating Function of GSTFSP
  13. Mean, variance and covariance for GSTFCP
  14. Special cases of GSTFCP
  15. Fractional order moments for GSTFCP and GSFCP
  16. ...and 13 more sections

Key Result

Proposition 3.1

The p.g.f. of GSTFSP for $|u|<1$ is given by

Figures (3)

  • Figure 1: Sample path of GSFSP for $\beta=0.4$ (black) and $\beta=0.7$ (green), $\lambda_1=1$, $\lambda_2=3$, $\lambda_3=2$, $\lambda_4=2$, $\lambda_5=2$, $\mu_1=2$, $\mu_2=2$, $\mu_3=3$, $\mu_4=3$, $\mu_5=2$.
  • Figure 2: Sample paths of GSTFSP for $\alpha=0.3$, $\beta=0.4$ (black); $\alpha=0.6$, $\beta=0.4$ (green); $\alpha=0.3$, $\beta=0.7$ (red); $\alpha=0.6$, $\beta=0.7$ (blue), $\lambda_1=1$, $\lambda_2=3$, $\lambda_3=2$, $\lambda_4=2$, $\lambda_5=2$, $\mu_1=2$, $\mu_2=2$, $\mu_3=3$, $\mu_4=3$, $\mu_5=2$.
  • Figure 3: p.m.f. of GSTFSP at times $t = 0.10$, $0.25$, $0.40$, $0.55$, $0.70$ for $\alpha=0.6$, $\beta=0.4$, $\Lambda=50$, $T=100$.

Theorems & Definitions (58)

  • Definition 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 48 more