The Fractional Poisson Process and the Inverse Stable Subordinator
28 Aug 2011-
TL;DR: In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.
Abstract: 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.
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TL;DR: In this paper, the authors developed a fractional calculus and theory of diffusion equations associated with operators in the time variable, where k is a nonnegative locally integrable function, and the solution of the Cauchy problem for the relaxation equation was proved (under some conditions upon k) continuous on [0, ∞) and completely monotone.
Abstract: We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form \({(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int
olimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}\) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation \({\mathbb D_{(k)} u=-\lambda u}\), 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. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.
283 citations
TL;DR: In this article, the correlation function for time-changed L evy processes has been studied in the context of continuous time random walks, where the second-order correlation function of a continuous-time random walk is defined.
Abstract: Time-changed L evy processes include the fractional Poisson process, and the scaling limit of a continuous time random walk. They are obtained by replacing the deterministic time variable by a positive non-decreasing random process. The use of time-changed processes in modeling often requires the knowledge of their second order properties such as the correlation function. This paper provides the explicit expression for the correlation function for time-changed L evy processes. The processes used to model random time include subordinators and inverse subordinators, and the time-changed L evy processes include limits of continuous time random walks. Several examples useful in applications are discussed.
68 citations
TL;DR: This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a temperamental telegraph equation.
Abstract: The inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements, with an exponential tempering that allows all moments to exist. This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a tempered time-fractional telegraph equation. Two explicit formulae for the density function are developed, and applied to compute explicit solutions to tempered fractional Cauchy problems, where a tempered fractional derivative replaces the first derivative in time. Several examples are given, including tempered fractional diffusion equations on bounded or unbounded domains, and the probability distribution of a tempered fractional Poisson process. It is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin.
57 citations
Cites background from "The Fractional Poisson Process and ..."
...7 in [55] shows that the pdfw(t) of the waiting times Jn has Laplace transform ∞...
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...1 in [55] shows that one can also write N(t) = max{n ≥ 0 : Tn ≤ t} (7....
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...2 in [55] shows that one can also construct the fractional Poisson process by replacing the time t in the traditional Poisson process N1(t)with an independent inverse stable subordinator....
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TL;DR: In this article, the authors considered time-changed Poisson processes and derived the governing difference-differential equations (DDEs) for these processes, and derived a new governing partial differential equation for the tempered stable subordinator of index 0 β 1.
51 citations
Cites background from "The Fractional Poisson Process and ..."
...Then the density qk(t) = P ( N(E(t)) = k ) solves (see Meerschaert et al. (2011)). dβ dtβ qk(t) = −λ(1−▽)qk(t)....
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...There is an interesting connection between continuous time random walks and fractional Cauchy problems, see [20, 18, 19] It is well known that the Poisson process N(t) with parameter λ > 0 solves the following difference-differential equation (DDE) d dt pk(t) = −λpk(t) + λpk−1(t), (1....
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...Also, the density of Jn is fJn(x) = g(x)e −axη + a β η , (2.12) with g(x) = d dx [1− Eβ(−ηxβ)] and η = λ− aβ (2.13) (see Example 5.7 of Meerschaert et al. (2011))....
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TL;DR: In this paper, the authors define fractional Skellam processes via the time changes in Poisson and Skekam processes by an inverse of a standard stable subordinator.
Abstract: The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.
47 citations
Cites background from "The Fractional Poisson Process and ..."
...[23] showed that the same fractional Poisson process can also be obtained via an inverse stable time change....
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...It is also proven in [23] that the definition of the fractional Poisson process as a renewal process with Mittag-Leffler distribution of inter-arrival times is equivalent to the time change definition Nα(t) = N1(E(t)), where N1(t), t ≥ 0 is a homogeneous Poisson process with parameter λ > 0 and E(t), t ≥ 0 is the inverse stable subordinator independent of N1(t)....
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References
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TL;DR: In this article, different fractional versions of the compound Poisson process are studied and the corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one.
Abstract: We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.
33 citations
TL;DR: This paper generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag–Leffler function in the kernel.
33 citations
TL;DR: In this article, the mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature.
Abstract: We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.
31 citations
TL;DR: A type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu's CSBP block-counting process arising in sampling from P D ( e - t , 0 ) .
31 citations
TL;DR: In this paper, a martingale characterization for the Fractional Poisson process on the plane is given, and the authors extend this result to the mixed-fractional poisson process and show that this process is the solution of a system of fractional differential-difference equations.
Abstract: We present new properties for the Fractional Poisson process and the Fractional
Poisson field on the plane. A martingale characterization for Fractional Poisson
processes is given. We extend this result to Fractional Poisson fields, obtaining some
other characterizations. The fractional differential equations are studied. We consider
a more general Mixed-Fractional Poisson process and show that this process is the
stochastic solution of a system of fractional differential-difference equations. Finally,
we give some simulations of the Fractional Poisson field on the plane.
30 citations