The Fractional Poisson Process and the Inverse Stable Subordinator

By Mark M. Meerschaert, Erkan Nane, Palaniappan Vellaisamy. 2010

260 citations recorded by the source corpus.

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/8de484ab7233384990fe1c0d3eb9c1f3b3a6bb5a

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Citations in the corpus

A practical guide to Prabhakar fractional calculus (2020, unknown): “[3, 4, 90, 103, 122] and references therein.”
Fractional Skellam processes with applications to finance (2014, background): “[23] showed that the same fractional Poisson process can also be obtained via an inverse stable time change.”
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 pαx(t) = P{Nλ(Yα(t)) = x} = ∫ ∞”
General Fractional Calculus, Evolution Equations, and Renewal Processes (2011, methodology): “The solution of the Cauchy problem for the relaxation equation D(k)u = −λu, λ > 0, proved to be (under some conditions upon k) continuous on [(0,∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process.”
Inverse Tempered Stable Subordinators (2014, methodology): “The inverse stable process defined by E(t) = inf{s > 0 : D(s) > t} has been widely used, as a time-change (see, Meerschaert et al. (2011); Hahn et al. (2011) and references therein).”
A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability (2017, background): “he following generalization, namely the fractional COM-Poisson distribution PfNn;a,g(t) = ng= (lt)n (G(an +g))n 1 En;a,g lt) , (23) where En;a,g(t) = ¥ å k=0 tk (G(ak +g))n , a > 0, n > 0, g 2R (24) is the a-Mittag–Lefﬂer function, considered by Gerhold [11], Garra and Polito [9]. The distribu”
A fractional counting process and its connection with the Poisson process (2015, background): “Since the transition density (2.13) coincides with the probability density function of the standard inverse ν-stable subordinator Eν(t) (see Meerschaert et al., 2011), the result given in Proposition 2.6 can be stated also as follows: The process Mν (t) and the process M1 (Eν(t)) are identically…”