Fractional Poisson Fields and Martingales

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.

References in corpus (4)

• A Generalization of the Space-Fractional Poisson Process and its Connection to some Lévy Processes
• Renewal processes of Mittag-Leffler and Wright type
• Correlation structure of time-changed fractional Brownian motion
• Finite Dimensional Fokker-Planck Equations for Continuous Time Random Walks

Cited by in corpus (7)

• Mixed fractional Risk Process
• Convoluted Fractional Poisson Process
• On the Long-Range Dependence of Mixed Fractional Poisson Process
• Properties of Poisson processes directed by compound Poisson-Gamma subordinators
• Mixtures of Tempered Stable Subordinators
• Fractional Skellam Process of Order $k$
• Some Time-changed fractional Poisson processes