# 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.

AbstractThe 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.

202 citations

Journal ArticleDOI
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.

55 citations

Journal ArticleDOI
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.

48 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  ∞...

[...]

• ...1 in [55] shows that one can also write N(t) = max{n ≥ 0 : Tn ≤ t} (7....

[...]

• ...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....

[...]

Journal ArticleDOI
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.
Abstract: We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDEs. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDEs corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index 0 β 1 , when β is a rational number. We then use this result to obtain the governing DDE for the mass function of the Poisson process time-changed by the tempered stable subordinator. Our results extend and complement the results in Baeumer et al. (2009) and Beghin and Orsingher (2009) in several directions.

47 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)....

[...]

• ...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....

[...]

• ...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))....

[...]

Journal ArticleDOI
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.

39 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....

[...]

• ...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)....

[...]

##### References
More filters
Journal ArticleDOI
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.

55 citations

Journal ArticleDOI
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.

48 citations

Journal ArticleDOI
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.
Abstract: We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDEs. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDEs corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index 0 β 1 , when β is a rational number. We then use this result to obtain the governing DDE for the mass function of the Poisson process time-changed by the tempered stable subordinator. Our results extend and complement the results in Baeumer et al. (2009) and Beghin and Orsingher (2009) in several directions.

47 citations

Journal ArticleDOI
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.

39 citations

Posted Content
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