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The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
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TLDR
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.read more
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Inverse Tempered Stable Subordinators
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.
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Large deviations for fractional Poisson processes
Luisa Beghin,Claudio Macci +1 more
TL;DR: In this article, the authors prove large deviation principles for two versions of fractional Poisson processes: the main version is a renewal process, the alternative version is weighted Poisson process.
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Alternative Forms of Compound Fractional Poisson Processes
Luisa Beghin,Claudio Macci +1 more
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.
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Renewal processes based on generalized Mittag–Leffler waiting times
Dexter O. Cahoy,Federico Polito +1 more
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.
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On the fractional Poisson process and the discretized stable subordinator
TL;DR: In this paper, the renewal counting number process N = N(t) is considered as a forward march over the non-negative integers with independent identically distributed waiting times, and the Laplace transform with respect to both variables x and t is applied.
References
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Journal ArticleDOI
Correlation Structure of Time-Changed Lévy Processes
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.
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Applications of inverse tempered stable subordinators
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.
Journal ArticleDOI
Time-changed Poisson processes
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.
Journal ArticleDOI
Fractional Skellam processes with applications to finance
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.
Posted Content
Inverse Tempered Stable Subordinators
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.