Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional Skellam Process of Order $k$

K. K. Kataria, M. Khandakar

Abstract

We introduce and study a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order $k$ (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of the FSPoK which are utilized to establish its long-range dependence property. Later, we considered two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.

Fractional Skellam Process of Order $k$

Abstract

We introduce and study a fractional version of the Skellam process of order by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of the FSPoK which are utilized to establish its long-range dependence property. Later, we considered two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.

Paper Structure

This paper contains 11 sections, 9 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Fractional Skellam process of order $k$
  3. The FSPoK time-changed by a Lévy subordinator
  4. Some special cases of the TCFSPoK
  5. FSPoK time-changed by gamma subordinator
  6. SPoK time-changed by tempered stable subordinator
  7. SPoK time-changed by inverse Gaussian subordinator
  8. The FSPoK time-changed by inverse subordinator
  9. SPoK time-changed by the inverse TSS
  10. SPoK time-changed by the first hitting time of IG subordinator
  11. Concluding remarks

Key Result

Proposition 2.1

The state probabilities $p_{\alpha}^{k}(n,t),\ n\in \mathbb{Z}$ of FSPoK solves the following system of fractional differential equations: with the initial conditions $p^{k}_{\alpha}(0,0)=1$ and $p^{k}_{\alpha}(n,0)=0, \ n\neq0$. Here, $\partial_{t}^{\alpha}$ denotes the Caputo fractional derivative defined as (see Kilbas et al. (2006))

Theorems & Definitions (22)

  • Proposition 2.1
  • proof
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.1
  • proof
  • proof
  • ...and 12 more