Fractional Poisson Fields and Martingales

Giacomo Aletti; Nikolai Leonenko; Ely Merzbach

doi:10.1007/s10955-018-1951-y

Fractional Poisson Fields and Martingales

Giacomo Aletti, Nikolai Leonenko, Ely Merzbach

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

31 Scopus citations

Abstract

We present new properties for the Fractional Poisson process (FPP) and the Fractional Poisson field on the plane. A martingale characterization for FPPs 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.

Original language	English
Pages (from-to)	700-730
Number of pages	31
Journal	Journal of Statistical Physics
Volume	170
Issue number	4
DOIs	https://doi.org/10.1007/s10955-018-1951-y
State	Published - 1 Feb 2018

Bibliographical note

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Funding

N. Leonenko was supported in particular by Cardiff Incoming Visiting Fellowship Scheme and International Collaboration Seedcorn Fund and Australian Research Council’s Discovery Projects funding scheme (Project No. DP160101366).

Funders	Funder number
Cardiff Incoming Visiting Fellowship Scheme and International Collaboration Seedcorn Fund
Australian Research Council

Keywords

Fractional Poisson fields
Fractional differential equations
Inverse subordinator
Martingale characterization
Second order statistics

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10.1007/s10955-018-1951-y

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AB - We present new properties for the Fractional Poisson process (FPP) and the Fractional Poisson field on the plane. A martingale characterization for FPPs 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.

KW - Fractional Poisson fields

KW - Fractional differential equations

KW - Inverse subordinator

KW - Martingale characterization

KW - Second order statistics

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Fractional Poisson Fields and Martingales

Abstract

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Keywords

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