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On some extensions of generalized counting processes

Lyudmyla Sakhno, Artem Storozhuk

Abstract

We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing equations for probability distributions and probability generating functions which involve fractional derivatives of different orders. Closed form expressions for probability distributions and probability generating functions are also provided for several considered models.

On some extensions of generalized counting processes

Abstract

We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing equations for probability distributions and probability generating functions which involve fractional derivatives of different orders. Closed form expressions for probability distributions and probability generating functions are also provided for several considered models.

Paper Structure

This paper contains 15 sections, 19 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized fractional derivatives
  4. Mittag-Leffler functions
  5. Fractional Poisson process
  6. Generalized fractional counting processes
  7. Sum of stable subordinators and its inverse and corresponding time-changed counting processes
  8. Equation for the density of the inverse process
  9. Fractional Poisson processes corresponding to time-change by the inverse to the sum of stable subordinators
  10. Generalized fractional counting processes corresponding to time-change by the inverse to the sum of stable subordinators
  11. Further generalizations
  12. Sum of tempered stable subordinators and its inverse and corresponding time-changed counting processes
  13. Equation for the density of the inverse process
  14. Generalized counting processes corresponding to time-change by the inverse to the sum of tempered stable subordinators
  15. An application to risk theory

Key Result

Theorem 1

For all fixed $\nu \in (0, 1]$ we have where $N_\Lambda^\nu(t)$ is a fractional Poisson process (see Section fractionalPoisson), with intensity $\Lambda = \lambda_1 + \lambda_2 + \dots + \lambda_k$, and $\{X_n : n \ge 1\}$ is a sequence of i.i.d. random variables, independent of $N_\Lambda^\nu(t)$, such that for any $n \in \mathbb{N}$ and where both $N_\Lambda^\nu(t)$ and $X_n$ depend on the same

Theorems & Definitions (36)

  • Theorem 1: DiC
  • Theorem 2: DiC
  • Theorem 3: BS2024
  • Theorem 4: DovOrsToaldo
  • Remark 1
  • Remark 2
  • Theorem 5
  • proof
  • Theorem 6: BO2010
  • Theorem 7
  • ...and 26 more