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.
Citations
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TL;DR: In this paper, an approach based on Levy mixing is proposed for nondecreasing Levy processes associated with Levy triplets of the form (a ( y ), b ( y ), ν ( d s, y ) ) and the parameter y is randomized by means of a probability measure.
18 citations
Cites background from "The Fractional Poisson Process and ..."
...[29] of the process N(L (t)), t > 0, where N is a Poisson process with rate λ and L is an inverse subordinator....
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TL;DR: In this paper, the convergence of a deterministic function driven by a time-changed symmetric α-stable Levy process is proved in the Skorokhod space endowed with the M1-topology of a sequence of stochastic integrals.
17 citations
Cites background from "The Fractional Poisson Process and ..."
...It has almost surely continuous non-decreasing sample paths and without stationary and independent increments (see [32])....
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TL;DR: In this article, the authors presented multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent (non-fractional) poisson processes.
Abstract: In this paper we present multivariate space-time fractional Poisson processes by considering
common random time-changes of a (finite-dimensional) vector of independent
classical (nonfractional) Poisson processes. In some cases we also consider compound
processes. We obtain some equations in terms of some suitable fractional derivatives and
fractional difference operators, which provides the extension of known equations for the
univariate processes.
16 citations
Cites background from "The Fractional Poisson Process and ..."
...[1] and [21]); for the inverse of stable subordinators, we recall [7], [13] and [17]....
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...In this paper we deal with fractional Poisson processes which are the main examples among counting processes; here we recall the references [11], [12], [4], [5], [15] and [19] (we also cite [10] and [13] where their representation in terms of randomly time-changed and subordinated processes was studied in detail)....
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TL;DR: In this paper, the authors defined a fractional negative binomial process (FNBP) by replacing the Poisson process by a FPP in the gamma subordinated form of the negative Binomial process.
Abstract: In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $\lambda$ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to $pde$'$s$ governing the density of the FNBP and the SFPP are also investigated.
16 citations
TL;DR: In this paper, the Riemann-Liouville fractional integral N α, ν ( t ) = 1 Γ ( α ) ∫ 0 t ( t − s ) α − 1 N ν( s ) d s, where N α, 1 ( t ), t ≥ 0, is a fractional Poisson process of order ν ∈ ( 0, 1 ], and α > 0.
15 citations
References
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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
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
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
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TL;DR: In this paper, the first-hitting time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator is considered, and the limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained.
Abstract: We consider the first-hitting time of a tempered $\beta$-stable subordinator, also called inverse tempered stable (ITS) subordinator. The density function of the ITS subordinator is obtained, for the index of stability $\beta \in (0,1)$. The series representation of the ITS density is also obtained, which could be helpful for computational purposes. The asymptotic behaviors of the $q$-th order moments of the ITS subordinator are investigated. In particular, the limiting behaviors of the mean of the ITS subordinator is given. The limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained. The governing PDE for the ITS density is also obtained. The corresponding known results for inverse stable subordinator follow as special cases.
35 citations