Fractional Poisson Process

By Nick Laskin. 2003

302 citations recorded by the source corpus.

Abstract A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process. As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1bd8f10cbc3e3e89a94d365db4962ca69cb9cfee

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Citations in the corpus

Fractional Skellam processes with applications to finance (2014, background): “[21, 22] studied the fractional Poisson process, where the exponential waiting time distribution is replaced by a Mittag-Leffler distribution, see also [4, 17, 28, 32].”
The fractional non-homogeneous Poisson process (2016, background): “It follows that for λ(t) = λ > 0, the FNPP coincides with the FHPP discussed by Meerschaert et al. (2011). In this case the marginal probabilities p x (t) = P{Nλ(Yα(t)) = x} = ∫ ∞”
A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability (2017, background): “s with fractional calculus, ﬁrst considered by Beghin and Orsingher in [3], is given by the fact that the probability generating function G(u,t) = ¥ å k=0 ukPfNa(t) = kg= Ea,1(ut) Ea,1(t) , a 2(0,1], (21) 5 of 10 satisﬁes the fractional differential equation daG(ua,t) dua = tG(ua,t), a 2(0,1], juj1,”
A fractional counting process and its connection with the Poisson process (2015, methodology): “A different approach has been developed by Laskin (2003) and Laskin (2009), where a fractional non-Markov Poisson stochastic process based on a fractional generalization of the Kolmogorov-Feller equations, and some interesting applications including a fractional compound Poisson process have been…”
The Fractional Poisson Process and the Inverse Stable Subordinator (2010, methodology): “(−λtβ)r Γ(β(r + n) + 1) which is the same form obtained by Jumarie [22], Laskin [27], Beghin and Orsingher [5, 6] and Cahoy [9] using different methods.”
Poisson-type processes governed by fractional and higher-order recursive differential equations (2009, methodology): “ Fractional versions of the Poisson processes have been already presented and studied in the literature: in [9] the so-called fractional master equation was considered. A similar model was treated in [12], where the equation governing the probability distribution of the homogeneous Poisson process w”
On a fractional linear birth-death process (2011, background): “Furthermore, the qualitative features illustrated in the last section show that the fractional counterpart of the linear birth–death process has a faster mean evolution (and variance expansion), as was pointed out in similar fractional generalizations, for example, for the Poisson process (see [2, 3”
Fractional Poisson processes and related planar random motions (2009, methodology): “1 Introduction Attempts to construct fractional versions of the Poisson process have been undertaken by Repin and Saichev (2000), Jumarie (2001) and Laskin (2003).”
Parameter estimation for fractional Poisson processes (2010, unknown): “From equation (23), we obtain μ = exp(−ν[E ln(T ) + C]).”
A practical guide to Prabhakar fractional calculus (2020, unknown): “[3, 4, 90, 103, 122] and references therein.”