We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order $\nu >0$. In the case $\nu >1$, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index $1/\nu $. For $\nu $ less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., \cite{VAN} and the references therein). As a consequence of the previous results, the fractional generalizations 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. These processes are particularly relevant for a wide range of financial and technological applications.

Fractional Gamma and Gamma-Subordinated Processes / Beghin, Luisa. - In: STOCHASTIC ANALYSIS AND APPLICATIONS. - ISSN 0736-2994. - STAMPA. - 33:5(2015), pp. 903-926. [10.1080/07362994.2015.1053615]

Fractional Gamma and Gamma-Subordinated Processes

BEGHIN, Luisa
2015

Abstract

We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order $\nu >0$. In the case $\nu >1$, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index $1/\nu $. For $\nu $ less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., \cite{VAN} and the references therein). As a consequence of the previous results, the fractional generalizations 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. These processes are particularly relevant for a wide range of financial and technological applications.
2015
fractional shift operator; gamma process; Mittag-Leffler function; stable subordinator; stochastic deterioration; time-changed processes; appliedmMathematics; statistics and probability; statistics, probability and uncertainty
01 Pubblicazione su rivista::01a Articolo in rivista
Fractional Gamma and Gamma-Subordinated Processes / Beghin, Luisa. - In: STOCHASTIC ANALYSIS AND APPLICATIONS. - ISSN 0736-2994. - STAMPA. - 33:5(2015), pp. 903-926. [10.1080/07362994.2015.1053615]
File allegati a questo prodotto
File Dimensione Formato  
Beghin_Fractional-gamma_2015.pdf

solo gestori archivio

Tipologia: Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 182.18 kB
Formato Adobe PDF
182.18 kB Adobe PDF   Contatta l'autore

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/865213
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 14
social impact