State-dependent Fractional Point Processes

TL;DR

This paper analyzes a state-dependent fractional Poisson process with variable fractional orders, providing explicit formulas, comparing different models, and introducing a new weighted sum process, advancing understanding of fractional stochastic processes.

Contribution

It introduces a novel state-dependent fractional Poisson process with explicit Laplace transforms and representations, and compares it to existing models, including a weighted sum approach.

Findings

Derived explicit Laplace transforms of state probabilities.

Identified differences between state-dependent and fractional Poisson processes.

Proposed a new weighted sum model of Poisson processes.

Abstract

The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{\nu_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $\nu_k$ differs from that constructed from the fractional state equations (in the case $\nu_k = \nu$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.