We study the connection between PDEs and L\'{e}vy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.

Fractional Poisson process with random drift / Beghin, Luisa; D'Ovidio, Mirko. - In: ELECTRONIC JOURNAL OF PROBABILITY. - ISSN 1083-6489. - ELETTRONICO. - 19, n.122(2014), pp. 1-26. [10.1214/EJP.v19-3258]

Fractional Poisson process with random drift

BEGHIN, Luisa;D'OVIDIO, MIRKO
2014

Abstract

We study the connection between PDEs and L\'{e}vy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11573/667811
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