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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06 Sep 2018
TL;DR: In this paper, a fractional M/M/1 queue with catastrophes is defined and studied, in particular focusing on the transient behaviour, in which the time-change plays a key role.
Abstract: Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al in 2003, we define and study a fractional M/M/1 queue with catastrophes In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe
23 citations
Cites background from "The Fractional Poisson Process and ..."
...We can conclude that, in the fractional case, catastrophes occur according to a fractional Poisson process ([1,17,20]) with rate ξ if the system is not empty....
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...a time-changed birth-death process [17,20], by means of an inverse stable subordinator [21], solves the ✷✽...
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TL;DR: In this article, the fractional Poisson process was generated by subordinating the standard Poisson processes to the inverse stable subordinator, based on application of the Laplace transform with respect to both arguments of the evolving probability densities.
Abstract: We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities.
21 citations
Cites background or result from "The Fractional Poisson Process and ..."
...Let us note that by solving the system (33), Beghin and Orsingher in [1] introduce what they call the ”first form of the fractional Poisson process” , and in [29] Meerschaert, Nane and Vellaisamy show that this process is a renewal process with Mittag-Leffler waiting time density as in (17), hence is identical with the ”fractional Poisson process”....
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...and Meerschaert, Nane and Vellaisamy [29]....
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...Remark According to Meerschaert, Nane and Vellaisamy [29] zβ(t) = z1(y(t∗(t))) , (41) where zβ(t) is a CTRW with β-Mittag-Leffler waiting times and an arbitrary jump density, and z1 is a CTRW with the exponential waiting times of the standard Poisson process having the same arbitrary jump distribution....
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...Beghin and Orsingher call the process so generated by subordination the ”first form of the fractional Poisson process” (in fact they consider other kinds of generalization, too) whereas the authors of [29] call it the ”fractal Poisson process” and show that it is a renewal process with the same waiting time distribution as the fractional Poisson process....
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...(40) Remark According to Meerschaert, Nane and Vellaisamy [29] zβ(t) = z1(y(t∗(t))) , (41) where zβ(t) is a CTRW with β-Mittag-Leffler waiting times and an arbitrary jump density, and z1 is a CTRW with the exponential waiting times of the standard Poisson process having the same arbitrary jump distribution....
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TL;DR: In this paper, a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and the related governing convolution type equations are obtained.
Abstract: In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.
21 citations
TL;DR: It is demonstrated that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime, and particular risk measures are addressed, which allow simple evaluations in an environment governed by the fractionAL Poisson process.
Abstract: The Poisson process suitably models the time of successive events and thus has numerous applications in statistics, in economics, it is also fundamental in queueing theory. Economic applications include trading and nowadays particularly high frequency trading. Of outstanding importance are applications in insurance, where arrival times of successive claims are of vital importance. It turns out, however, that real data do not always support the genuine Poisson process. This has lead to variants and augmentations such as time dependent and varying intensities, for example. This paper investigates the fractional Poisson process. We introduce the process and elaborate its main characteristics. The exemplary application considered here is the Carmer–Lundberg theory and the Sparre Andersen model. The fractional regime leads to initial economic stress. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process.
20 citations
Cites background from "The Fractional Poisson Process and ..."
...[28] find the following stochastic representation for FPP: Nα(t) = N ( Yα(t) ) , t ≥ 0, α ∈ (0, 1), where N = {N(t), t ≥ 0}, is the classical homogeneous Poisson process with parameter λ > 0, which is independent of the inverse stable subordinator Yα....
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TL;DR: In this article, the connection between PDEs and Levy processes running with clocks given by time-changed Poisson processes with stochastic drifts was studied, and the transition probability laws were explicitly represented by the translation operator associated to the representation of Poisson semigroup.
Abstract: We study the connection between PDEs and Levy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.
19 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