Electronic Journal of Probability

Fractional Poisson process with random drift

Luisa Beghin and Mirko D'Ovidio

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We study the connection between PDEs and Lévy 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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 122, 26 pp.

Accepted: 30 December 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
Secondary: 60G50: Sums of independent random variables; random walks

Poisson process time-change random drift fractional equation Poisson semigroup

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Beghin, Luisa; D'Ovidio, Mirko. Fractional Poisson process with random drift. Electron. J. Probab. 19 (2014), paper no. 122, 26 pp. doi:10.1214/EJP.v19-3258. https://projecteuclid.org/euclid.ejp/1465065764

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