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Journal ArticleDOI

The space-fractional Poisson process

01 Apr 2012-Statistics & Probability Letters (North-Holland)-Vol. 82, Iss: 4, pp 852-858
TL;DR: In this paper, the authors introduce the space-fractional Poisson process whose state probabilities p, t, t > 0, � 2 (0,1), are governed by the equations (d/dt)pk(t) = � � (1 B)p � (t), where (B) is the fractional difference operator found in the study of time series analysis.
About: This article is published in Statistics & Probability Letters.The article was published on 2012-04-01 and is currently open access. It has received 110 citations till now. The article focuses on the topics: Fractional Poisson process & Compound Poisson process.
Citations
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Journal ArticleDOI
TL;DR: A generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals is presented, which shows some applications in classical equations of mathematical physics such as the heat and the free electron laser equations.

196 citations


Cites background from "The space-fractional Poisson proces..."

  • ...Let us here only recall the few relevant papers [10, 11, 40, 4, 3, 5, 20, 29, 32, 28] and see also the references cited therein....

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Journal ArticleDOI
TL;DR: The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution.
Abstract: The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a Levy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These exact analytical results are compared to Monte Carlo simulations.

50 citations

Journal ArticleDOI
TL;DR: In this paper, two fractional versions of a family of nonnegative integer-valued processes are considered and it is shown that their probability mass functions solve fractional Kolmogorov forward equations.
Abstract: We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters

48 citations


Additional excerpts

  • ...We recall that this process was presented in [25] by referring to the fractional difference operator (1 − B) , for α ∈ (0, 1], and it is called a space-fractional Poisson process....

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  • ...[21] for the case ν ∈ (0, 1) and [25] for the case ν ∈ (1, ∞))....

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Journal ArticleDOI
TL;DR: In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.
Abstract: In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.

45 citations

Journal ArticleDOI
TL;DR: In this article, a generalization of the space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form is introduced.
Abstract: This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.

41 citations


Cites background or methods from "The space-fractional Poisson proces..."

  • ...Note that for γ = 0, δ = 1, the generating function (3.37) reduces to that of the space-time fractional Poisson process [Orsingher and Polito, 2012], while for γ = 0, ν = 1, δ ∈ (0,1), we obtain that of the tempered space-time fractional Poisson process....

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  • ...An interesting special case related to these equations is that regarding the so-called space-fractional Poisson process [Orsingher and Polito, 2012]....

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  • ...We saw from the above analysis (see Remark 2.3 and the subordination result) that the difference operator (3.24) is in practice connected with the Prabhakar derivative....

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  • ...Note that these generalize the difference-differential equations governing the state probabilities of a space-fractional Poisson process [Orsingher and Polito, 2012] i.e. d dt pk(t) = −λ α(1− B)αpk(t), pk(0) = δk,0, k ≥ 0, t ≥ 0, (3.6) and of course that of the classical homogeneous process for α→…...

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  • ...The Laplace transform of the process ηV ν ,δ t , t ≥ 0, can be written as Eexp −µ ηV ν ,δ t = exp −t h η+µν δ −ηδ i , t ≥ 0, µ > 0....

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References
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Book
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index

11,492 citations


Additional excerpts

  • ...1 Introduction Fractional Poisson processes studied so far have been obtained either by considering renewal processes with intertimes between events represented by Mittag–Leffler distributions [Mainardi et al., 2004, Beghin and Orsingher, 2009] or by replacing the time derivative in the equations governing the state probabilities with the fractional derivative in the sense of Caputo....

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  • ...By means of Laplace transforms, some simple manipulations lead to νGα(u, t) = Eν(−λ α tν(1− u)α), |u| ≤ 1, (2.27) where Eν(x) is the Mittag–Leffler function [Kilbas et al., 2006]....

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  • ...The space-fractional Poisson process has probability generating function Gα(u, t) = Eu Nα(t) = e−λ α(1−u)α t , |u| ≤ 1, (1.4) and can be compared with its time-fractional counterpart νG(u, t) = Eu Nν (t) = Eν (−λ(1− u)t ν) , |u| ≤ 1, (1.5) where Eν(x) = ∞ ∑ r=0 x r Γ(ν r + 1) , ν > 0, (1.6) is the one-parameter Mittag–Leffler function....

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  • ...rivative operator (see Kilbas et al. [2006]). By means of Laplace transforms, some simple manipulations lead to νGα(u,t) = Eν(−λ αtν(1−u)α), |u| ≤ 1, (2.27) where Eν(x) is the Mittag–Leffler function [Kilbas et al., 2006]. In turn, by expanding the above probability generating function we have that pα,ν k (t)= (−1)k k! X∞ r=0 (−λαtν)r Γ(ν r +1 ) Γ(αr +1) α − , k ≥ 0, α∈ (0,1], ν∈ (0,1]. (2.28) When α= 1 these probabi...

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Book
Ruey S. Tsay1
15 Oct 2001
TL;DR: The author explains how the Markov Chain Monte Carlo Methods with Applications and Principal Component Analysis and Factor Models changed the way that conventional Monte Carlo methods were applied to time series analysis.
Abstract: Preface. Preface to First Edition. 1. Financial Time Series and Their Characteristics. 2. Linear Time Series Analysis and Its Applications. 3. Conditional Heteroscedastic Models. 4. Nonlinear Models and Their Applications. 5. High-Frequency Data Analysis and Market Microstructure. 6. Continuous-Time Models and Their Applications. 7. Extreme Values, Quantile Estimation, and Value at Risk. 8. Multivariate Time Series Analysis and Its Applications. 9. Principal Component Analysis and Factor Models. 10. Multivariate Volatility Models and Their Applications. 11. State-Space Models and Kalman Filter. 12. Markov Chain Monte Carlo Methods with Applications. Index.

2,766 citations


"The space-fractional Poisson proces..." refers methods in this paper

  • .... In this paper we introduce a space-fractional Poisson process by means of the fractional difference operator ∆α= (1−B)α, α∈ (0,1], (1.1) which often appears in the study of long memory time series [Tsay, 2005]. The operator (1.1) implies a dependence of the state probabilities pα k (t) from all probabilities pα j (t), j <k. For α= 1 we recover the classical homogeneous Poisson process and the state pro...

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Journal ArticleDOI
TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.

302 citations

Journal ArticleDOI
TL;DR: In this article, 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, which unifies the two main approaches in stochastic theory of time-fractional diffusion equations.
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.

243 citations

Journal ArticleDOI
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
Abstract: We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $ u\in(0,1]$. For this process, denoted by $\mathcal{N}_ u(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_ u(t)= N(\mathcal{T}_{2 u}(t)),$ $t>0$. The time argument $\mathcal{T}_{2 u }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_ u.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $ u\in(0,1]$ we show that the random position has a Brownian behavior (for $ u =1/2$) or a cylindrical-wave structure (for $ u =1$).

233 citations


"The space-fractional Poisson proces..." refers background or methods in this paper

  • ...…(1.10) is similar to the following representation of the time-fractional Poisson process: Nν(t) d = N T2ν(t) , (1.11) where T2ν(t), t > 0, is a process whose one-dimensional distribution is obtained by folding the solution to the time-fractional diffusion equation [Beghin and Orsingher, 2009]....

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  • .... 1 Introduction FractionalPoissonprocessesstudiedsofarhavebeenobtained eitherbyconsidering renewal processeswith intertimesbetweeneventsrepresented byMittag–Lefflerdistributions [Mainardi et al.,2004,Beghin and Orsingher, 2009]or by replacing the time derivative in the equations governing the state probabilities with the fractional derivative in the sense of Caputo. In this paper we introduce a space-fractional Poisson pro...

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  • ...ional Poisson process: Nν(t) =d N T2ν(t) , (1.11) where T2ν(t), t >0, is a process whose one-dimensional distribution is obtained by folding the solution to the time-fractional diffusion equation [Beghin and Orsingher, 2009]. Finally we can note that the probability generating function (1.7), for all u ∈ (0,1), can be represented as νGα(u,t)= Pr ˆ min 0≤k≤Nν(t) X1/α k ≥ 1−u ˙ , (1.12) where the X ks are i.i.d. uniformly...

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  • ...…obtained either by considering renewal processes with intertimes between events represented by Mittag–Leffler distributions [Mainardi et al., 2004, Beghin and Orsingher, 2009] or by replacing the time derivative in the equations governing the state probabilities with the fractional derivative in…...

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