Open Access
The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
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
Citations
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On Space-Time Fractional Heat Type Non-Homogeneous Time-Fractional Poisson Equation
TL;DR: In this paper, the authors considered a space-time fractional heat equation with Riemann-Liouville derivative of non-homogeneous time-fractional Poisson process and showed that the solution grows exponentially for some small time interval.
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A Functional Limit Theorem for stochastic integrals driven by a time-changed symmetric \alpha-stable L\'evy process
Enrico Scalas,Noèlia Viles +1 more
TL;DR: In this paper, the convergence of a deterministic function driven by a time-changed symmetric \alpha-stable L\'evy process is proved in the Skorokhod space.
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Fractional statistical dynamics and fractional kinetics
TL;DR: In this paper, the authors apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies, and obtain the evolution of the density of particles in the fractional kinetics.
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Counting processes with Bern\v{s}tein intertimes and random jumps
Enzo Orsingher,Bruno Toaldo +1 more
TL;DR: In this paper, the authors considered point processes with independent increments and integer-valued jumps whose distribution is expressed in terms of Bern\v{s}tein functions with L\'evy measure.
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Correlated fractional counting processes on a finite time interval
TL;DR: Some correlated fractional counting processes on a finite time interval are presented and, when the correlation parameter is equal to zero, the univariate distributions coincide with the ones of the space-time fractional Poisson process in Orsingher and Polito (2012).
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.
Journal ArticleDOI
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.
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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.