Fractional Gamma process and fractional Gamma-subordinated processes

2013/05/08 by Luisa Beghin, Beghin, Luisa
Mathematics · #26A33 #33E12 #34A08 #60G52 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:26A33 #msc:33E12 #msc:34A08 #msc:60G52

paper · pdf · doi:10.48550/arxiv.1305.1753

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arxiv created 2013/05/08 · arxiv updated 2013/05/09

Abstract

We define and study fractional versions of the well-known Gamma subordinator Γ:=\Γ(t), t≥ 0\, which are obtained by time-changing % Γ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained.

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