Fractional Poisson Fields and Martingales

TL;DR

This paper explores new properties and characterizations of fractional Poisson processes and fields, including martingale properties, fractional differential equations, and simulations, advancing understanding of these stochastic models.

Contribution

It introduces martingale characterizations for fractional Poisson processes and extends these to fractional Poisson fields, including solutions to fractional differential equations.

Findings

Martingale characterization for fractional Poisson processes

Extension to fractional Poisson fields with new properties

Simulation results of fractional Poisson fields on the plane

Abstract

We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.